{"id":26647,"date":"2023-02-01T17:50:38","date_gmt":"2023-02-01T17:50:38","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/blog\/?p=26647"},"modified":"2025-02-03T09:20:06","modified_gmt":"2025-02-03T09:20:06","slug":"factor-theorem","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/blog\/factor-theorem\/","title":{"rendered":"Factor Theorem: All You Need to Know"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"26647\" class=\"elementor elementor-26647\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<div class=\"elementor-inner\">\n\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-2708efa elementor-section-stretched elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"2708efa\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-365e51b\" data-id=\"365e51b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4e4dc8b elementor-widget elementor-widget-image\" data-id=\"4e4dc8b\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-image\">\n\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1920\" height=\"500\" src=\"https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem.webp\" class=\"attachment-full size-full wp-image-26649\" alt=\"Factor Theorem\" loading=\"lazy\" srcset=\"https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem.webp 1920w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem-300x78.webp 300w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem-1024x267.webp 1024w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem-768x200.webp 768w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/02\/factor-theorem-1536x400.webp 1536w\" sizes=\"(max-width: 1920px) 100vw, 1920px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c1a6c9a elementor-section-boxed 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src=\"https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2025\/02\/factor-theorem.webp\" class=\"attachment-large size-large wp-image-83468\" alt=\"Factor Theorem\" loading=\"lazy\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ee9d77c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ee9d77c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4cc4f8d\" data-id=\"4cc4f8d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e8b0d0 elementor-widget elementor-widget-text-editor\" data-id=\"7e8b0d0\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p style=\"text-align: center;\"><em><span style=\"color: #000000;\"><strong>Update:<\/strong> This article was last updated on<\/span> <strong><span style=\"text-decoration: underline; color: #800080;\">3rd February 2025<\/span> <\/strong><span style=\"color: #000000;\">to reflect the accuracy and up-to-date information on the page.<\/span><\/em><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ea5fad2 elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ea5fad2\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6a66966\" data-id=\"6a66966\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c7b6d5 elementor-author-box--layout-image-above elementor-author-box--align-center elementor-widget elementor-widget-author-box\" 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data-widget_type=\"post-info.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<ul class=\"elementor-inline-items elementor-icon-list-items elementor-post-info\">\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-ba9c996 elementor-inline-item\" itemprop=\"datePublished\">\n\t\t\t\t\t\t<a href=\"https:\/\/mp.moonpreneur.com\/blog\/2023\/02\/01\/\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-date\">\n\t\t\t\t\t\t\t\t\t\tFebruary 1, 2023\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t<\/li>\n\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-afbda51 elementor-inline-item\" itemprop=\"about\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-terms\">\n\t\t\t\t\t\t\t<span class=\"elementor-post-info__item-prefix\">,<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-post-info__terms-list\">\n\t\t\t\t<a href=\"https:\/\/mp.moonpreneur.com\/blog\/category\/advance-math\/\" class=\"elementor-post-info__terms-list-item\">Advance Math<\/a>\t\t\t\t<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<\/ul>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-inner-section elementor-element elementor-element-37d4c10 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"37d4c10\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-5e31761\" data-id=\"5e31761\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3dcc8da elementor-widget elementor-widget-text-editor\" data-id=\"3dcc8da\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">Factoring equations can be tricky and challenging, especially when your child needs to learn all the methods to solve them \u2013 and let\u2019s face it, memorizing long formulas isn\u2019t often their priority.\u00a0<\/span><\/p><p><a href=\"https:\/\/moonpreneur.com\/math-classes\/\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-large wp-image-25219\" src=\"https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner-1024x155.jpg\" alt=\"Advanced Math\" width=\"1024\" height=\"155\" srcset=\"https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner-1024x155.jpg 1024w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner-300x45.jpg 300w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner-768x116.jpg 768w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner-1536x232.jpg 1536w, https:\/\/mp.moonpreneur.com\/blog\/wp-content\/uploads\/2023\/01\/advanced-math-banner.jpg 2023w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p><p><span style=\"color: #000000;\">But don\u2019t worry! With the Factor Theorem, you\u2019ll easily understand how best to approach these complex problems. In this blog post, we\u2019ll discuss what exactly the Factor Theorem has to offer and why it is necessary for simplifying equations so that you and your child can gain a better understanding of this crucial mathematical concept. <\/span><\/p><h2 style=\"text-align: center;\"><span style=\"color: #003300;\"><b>What is a Factor Theorem?<\/b><\/span><\/h2><p><span style=\"color: #000000;\">The factor theorem states that if a polynomial p(x) can be divided by (x-a) with no remainder, then p(a) = 0. Conversely, if p(a) = 0, then (x-a) is a factor of p(x). Here, the Factor theorem provides a way to find the roots, or x-values, that make p(x) equal to zero.<\/span><\/p><h3 style=\"text-align: center;\"><span style=\"color: #003300;\"><strong>Proof of Factor Theorem:<\/strong><\/span><\/h3><p><span style=\"color: #000000;\">The proof of the factor theorem involves showing that if p(x) is divisible by (x-a), then p(a) = 0, and conversely, if p(a) = 0, then (x-a) is a factor of p(x).<\/span><\/p><p><em><span style=\"color: #333300;\"><strong>To show that if p(x) is divisible by (x-a), then p(a) = 0, consider the polynomial division:<\/strong><\/span><\/em><\/p><p><span style=\"font-weight: 400;\">p(x) = (x-a) q(x) + r, where q(x) and r are polynomials, and the degree of r is less than the degree of (x-a), which is 1.<\/span><\/p><p><span style=\"font-weight: 400;\">Setting x = a in the equation above, we get:<\/span><\/p><p><span style=\"font-weight: 400;\">p(a) = (a-a) q(a) + r = r.<\/span><\/p><p><span style=\"font-weight: 400;\">Since the degree of r is less than the degree of (x-a), we have r = 0. Hence, p(a) = 0.<\/span><\/p><p><span style=\"font-weight: 400;\">To show that if p(a) = 0, then (x-a) is a factor of p(x), consider the polynomial:<\/span><\/p><p><span style=\"font-weight: 400;\">p(x) \u2013 p(a) = (x-a) q(x) for some polynomial q(x).<\/span><\/p><p><span style=\"font-weight: 400;\">Since p(a) = 0, we have:<\/span><\/p><p><span style=\"font-weight: 400;\">p(x) = (x-a) q(x).<\/span><\/p><p><span style=\"font-weight: 400;\">This shows that (x-a) is a factor of p(x).<\/span><\/p><p><span style=\"color: #333300;\"><em><strong>Therefore, the factor theorem is proven.<\/strong><\/em><\/span><\/p><p><em><b>Example of Factor Theorem: Factorize y2 \u2013 5y + 6<\/b><\/em><\/p><p><span style=\"font-weight: 400;\">Let p(y) = y2 \u2013 5y + 6.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">p(y) = (y \u2013 a)(y \u2013 b)<\/span><\/p><p><span style=\"font-weight: 400;\">= y2 \u2013 by \u2013 ay + ab<\/span><\/p><p><span style=\"font-weight: 400;\">On comparing the constants, we get ab = 6.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Next, the factors of 6 are 1, 2, and 3.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Now, p(2) = 22\u2013 (5 \u00d7 2) + 6 = 4 \u2013 10 + 6 = 0.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">So, (y \u2013 2) is a factor of p(y).\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Also, p(3) = 32 \u2013 (5 \u00d7 3) + 6 = 9 \u2013 15 + 6 = 0.<\/span><\/p><p><span style=\"font-weight: 400;\">\u00a0So, (y \u2013 3) is also a factor of y2 \u2013 5y + 6.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, y2 \u2013 5y + 6 = (y \u2013 2)(y \u2013 3)<\/span><\/p><p style=\"text-align: left;\"><strong><span style=\"color: #000000;\">Recommended Reading: <\/span><\/strong><strong><a href=\"https:\/\/mp.moonpreneur.com\/blog\/deeper-understanding-of-math-concepts\/\">Developing a Deeper Understanding of Math Concepts<\/a><\/strong><\/p><h3 style=\"text-align: center;\"><span style=\"color: #003300;\"><b>Proof by Remainder Theorem:<\/b><\/span><\/h3><p>We can also prove the factor theorem using the remainder theorem. The remainder theorem states that for a polynomial p(x) and a constant a, the remainder when p(x) is divided by (x-a) is equal to p(a).<\/p><p>Using this result, we can prove the factor theorem as follows:<\/p><p>If p(x) is divisible by (x-a), then the remainder when p(x) is divided by (x-a) is 0. Therefore, p(a) = 0.<\/p><p>Conversely, if p(a) = 0, then the remainder when p(x) is divided by (x-a) is 0. This means that (x-a) is a factor of p(x).<\/p><p>Therefore, the factor theorem is proven using the remainder theorem.<\/p><p><strong>Recommended Reading: <a href=\"https:\/\/mp.moonpreneur.com\/blog\/vedic-math-for-kids\/\">Vedic Math for Kids: Why It Is Important<\/a><\/strong><\/p><h4 style=\"text-align: center;\"><span style=\"color: #003300;\"><b>How to Use Factor Theorem\u00a0<\/b><\/span><\/h4><p><span style=\"font-weight: 400;\"><strong>Write the polynomial:<\/strong> Write the polynomial that you want to find the roots of or the polynomial that you want to factor.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Choose a value of \u201ca\u201d:<\/strong> Choose a constant \u201ca\u201d to test as a possible root of the polynomial.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Evaluate p(a):<\/strong> Substitute the value of \u201ca\u201d into the polynomial and evaluate the expression to get p(a).<\/span><\/p><p><span style=\"font-weight: 400;\">Check if p(a) = 0: If p(a) = 0, then (x-a) is a factor of the polynomial.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Factor the polynomial:<\/strong> If (x-a) is a factor of the polynomial, you can use polynomial division to factor the polynomial. Divide the polynomial by (x-a) and write the result in the form p(x) = (x-a) q(x) + r, where q(x) is the quotient and r is the remainder. If r = 0, then (x-a) is a factor of the polynomial with no remainder.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Repeat steps 2-5:<\/strong> Repeat steps 2-5 with different values of \u201ca\u201d until you have found all the roots of the polynomial.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Construct the factors:<\/strong> Use the roots that you found in step 6 to construct the factors of the polynomial. The polynomial can be written as the product of the factors.<\/span><\/p><p>In summary, the factor theorem provides a way to find the roots of a polynomial and to factor the polynomial. The process involves evaluating the polynomial at a constant \u201ca,\u201d checking if the result equals zero, and if so, using polynomial division to factor the polynomial.<\/p><p><b>Recommended Reading: <\/b><a href=\"https:\/\/mp.moonpreneur.com\/blog\/math-projects-ideas-for-kids\/\"><b>Top 5 Math Project Ideas for Kids<\/b><\/a><\/p><h5 style=\"text-align: center;\"><span style=\"color: #003300;\"><b>Why Do We Use Factor Theorem<\/b><\/span><\/h5><p><span style=\"color: #333300;\"><em><span style=\"font-weight: 400;\">The factor theorem is used for several reasons:<\/span><\/em><\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Finding roots of polynomials:<\/strong> The factor theorem provides a method for finding the roots, or x-values, that make a polynomial equal to zero.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Factoring polynomials:<\/strong> The factor theorem can be used to factor polynomials by finding the roots of the polynomial and then constructing the corresponding factors.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Simplifying polynomials:<\/strong> Through<\/span><span style=\"font-weight: 400;\"> factoring polynomials <\/span><span style=\"font-weight: 400;\">by<\/span><span style=\"font-weight: 400;\"> the factor theorem, <\/span><span style=\"font-weight: 400;\">one<\/span> <span style=\"font-weight: 400;\">can<\/span> <span style=\"font-weight: 400;\">reduce<\/span> <span style=\"font-weight: 400;\">the expression and make it easier to manipulate.<\/span><\/p><p><span style=\"font-weight: 400;\"><strong>Solving polynomial equations:<\/strong> The factor theorem can be employed to solve polynomial equations by<\/span><span style=\"font-weight: 400;\"> factoring the polynomial <\/span><span style=\"font-weight: 400;\">from the equation <\/span><span style=\"font-weight: 400;\">on one side and <\/span><span style=\"font-weight: 400;\">finding<\/span><span style=\"font-weight: 400;\"> the roots <\/span><span style=\"font-weight: 400;\">for<\/span><span style=\"font-weight: 400;\"> the solution.<\/span><\/p><p><span style=\"font-weight: 400;\">In <\/span><span style=\"font-weight: 400;\">other words, a factor theorem is one<\/span> <span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">the<\/span><span style=\"font-weight: 400;\"> algebra <\/span><span style=\"font-weight: 400;\">tools which may be applied for finding the roots of polynomials, factoring polynomials, simplifying expressions, and solving polynomial equations.<\/span><\/p><p><strong><span style=\"color: #000000;\">Recommended Reading: <\/span><a href=\"https:\/\/mp.moonpreneur.com\/blog\/is-advanced-calculus-the-hardest\/\">Is Advanced Calculus The Hardest Math Class In High School? A Student\u2019s Perspective<\/a><\/strong><\/p><h4 style=\"text-align: center;\"><span style=\"color: #003300;\"><b>What are the other methods to find the factors of polynomials?<\/b><\/span><\/h4><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Synthetic Division: It<\/span><span style=\"font-weight: 400;\"> is a <\/span><span style=\"font-weight: 400;\">division <\/span><span style=\"font-weight: 400;\">method <\/span><span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">a polynomial by a linear factor<\/span><span style=\"font-weight: 400;\"> which <\/span><span style=\"font-weight: 400;\">is<\/span><span style=\"font-weight: 400;\"> used <\/span><span style=\"font-weight: 400;\">in order to find the factors of the polynomial.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Long Division: This is one<\/span> <span style=\"font-weight: 400;\">way<\/span> <span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">doing<\/span> <span style=\"font-weight: 400;\">division<\/span> <span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">polynomials <\/span><span style=\"font-weight: 400;\">by <\/span><span style=\"font-weight: 400;\">polynomials.<\/span> <span style=\"font-weight: 400;\">These<\/span> <span style=\"font-weight: 400;\">are<\/span><span style=\"font-weight: 400;\"> used <\/span><span style=\"font-weight: 400;\">in<\/span> <span style=\"font-weight: 400;\">finding<\/span><span style=\"font-weight: 400;\"> the factors of <\/span><span style=\"font-weight: 400;\">a polynomial.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Rational Root Theorem: This is a <\/span><span style=\"font-weight: 400;\">theorem <\/span><span style=\"font-weight: 400;\">which<\/span> <span style=\"font-weight: 400;\">says <\/span><span style=\"font-weight: 400;\">that <\/span><span style=\"font-weight: 400;\">when a polynomial has a rational root, then such<\/span> <span style=\"font-weight: 400;\">a <\/span><span style=\"font-weight: 400;\">root <\/span><span style=\"font-weight: 400;\">has<\/span> <span style=\"font-weight: 400;\">to <\/span><span style=\"font-weight: 400;\">divide <\/span><span style=\"font-weight: 400;\">both the leading coefficient and the constant term of the polynomial. A<\/span> <span style=\"font-weight: 400;\">good <\/span><span style=\"font-weight: 400;\">theorem <\/span><span style=\"font-weight: 400;\">which<\/span> <span style=\"font-weight: 400;\">limits<\/span> <span style=\"font-weight: 400;\">the<\/span> <span style=\"font-weight: 400;\">list<\/span> <span style=\"font-weight: 400;\">of<\/span> <span style=\"font-weight: 400;\">possible<\/span> <span style=\"font-weight: 400;\">candidates<\/span><span style=\"font-weight: 400;\"> for roots.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Descartes<\/span><span style=\"font-weight: 400;\">&#8216;<\/span><span style=\"font-weight: 400;\"> Rule of Signs: <\/span><span style=\"font-weight: 400;\">According to <\/span><span style=\"font-weight: 400;\">this rule<\/span><span style=\"font-weight: 400;\">,<\/span> <span style=\"font-weight: 400;\">the number of positive roots of a polynomial is less than or equal to the number of sign changes in the coefficients of the polynomial.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Factoring by Grouping: This is factoring a polynomial by grouping its <\/span><span style=\"font-weight: 400;\">terms together<\/span><span style=\"font-weight: 400;\">,<\/span> <span style=\"font-weight: 400;\">then using the distributive property to factor <\/span><span style=\"font-weight: 400;\">terms.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Special Factoring Techniques<\/span><span style=\"font-weight: 400;\"> There are several special factoring techniques.<\/span> <span style=\"font-weight: 400;\">These<\/span> <span style=\"font-weight: 400;\">include<\/span><span style=\"font-weight: 400;\"> substitution, grouping, and <\/span><span style=\"font-weight: 400;\">sum and difference of cubes.<\/span><\/li><\/ul><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">In<\/span> <span style=\"font-weight: 400;\">a nutshell, the Factor Theorem is a very<\/span> <span style=\"font-weight: 400;\">important <\/span><span style=\"font-weight: 400;\">mathematical concept that <\/span><span style=\"font-weight: 400;\">will<\/span> <span style=\"font-weight: 400;\">help <\/span><span style=\"font-weight: 400;\">find the <\/span><span style=\"font-weight: 400;\">roots of polynomials and factor them.<\/span><\/span><\/p><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The Factor Theorem can be proven by polynomial division or the remainder theorem. But,<\/span> <span style=\"font-weight: 400;\">due<\/span> <span style=\"font-weight: 400;\">to<\/span> <span style=\"font-weight: 400;\">its effectiveness in solving complex polynomial problems, the Factor Theorem is very commonly applied<\/span><\/span><\/p><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Want to <\/span><span style=\"font-weight: 400;\">get<\/span><span style=\"font-weight: 400;\"> your child <\/span><span style=\"font-weight: 400;\">excited <\/span><span style=\"font-weight: 400;\">about math and <\/span><span style=\"font-weight: 400;\">sharpen<\/span> <span style=\"font-weight: 400;\">his<\/span><span style=\"font-weight: 400;\"> math skills?\u00a0\u00a0<\/span><\/span><\/p><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Moonpreneur<\/span><span style=\"font-weight: 400;\">&#8216;s online math curriculum is unique since<\/span><span style=\"font-weight: 400;\"> it <\/span><span style=\"font-weight: 400;\">lets<\/span><span style=\"font-weight: 400;\"> children understand <\/span><span style=\"font-weight: 400;\">their math skills through hands-on lessons, assists them in building real-life applications<\/span><span style=\"font-weight: 400;\"> and excites them <\/span><span style=\"font-weight: 400;\">toward<\/span> <span style=\"font-weight: 400;\">learning<\/span><span style=\"font-weight: 400;\"> math. You can <\/span><span style=\"font-weight: 400;\">choose<\/span> <span style=\"font-weight: 400;\">our <\/span><\/span><a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\"><span style=\"font-weight: 400;\">Advanced Math<\/span><\/a><span style=\"font-weight: 400; color: #000000;\"> or Vedic Math+Mental Math courses. 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updated on 3rd February 2025 to reflect the accuracy and up-to-date information on the page. Factoring equations can be tricky and challenging, especially when your child needs to learn all the methods to solve them \u2013 and let\u2019s face it, memorizing long formulas isn\u2019t often their priority.\u00a0 But don\u2019t worry! [&hellip;]<\/p>\n","protected":false},"author":825,"featured_media":26649,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[969],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/posts\/26647"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/users\/825"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/comments?post=26647"}],"version-history":[{"count":0,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/posts\/26647\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/media\/26649"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/media?parent=26647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/categories?post=26647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/blog\/wp-json\/wp\/v2\/tags?post=26647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}