{"id":225795,"date":"2025-06-04T15:40:05","date_gmt":"2025-06-04T15:40:05","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225795"},"modified":"2025-06-04T15:40:08","modified_gmt":"2025-06-04T15:40:08","slug":"what-is-the-factored-form-of-2x3-4x2-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/what-is-the-factored-form-of-2x3-4x2-x\/","title":{"rendered":"What is the factored form of 2×3 + 4×2 \u2013 x"},"content":{"rendered":"\n

What is the factored form of 2×3 + 4×2 \u2013 x?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The factored form of the expression 2×3+4×2\u2212x2x^3 + 4x^2 – x is: x(2x+1)(x\u22121)x(2x + 1)(x – 1)<\/p>\n\n\n\n

\n\n\n\n

Step-by-Step Explanation<\/strong><\/h3>\n\n\n\n

To factor the polynomial 2×3+4×2\u2212x2x^3 + 4x^2 – x, we follow a systematic approach using factoring techniques such as factoring out the greatest common factor (GCF), grouping, and factoring trinomials.<\/p>\n\n\n\n

Step 1: Factor out the Greatest Common Factor (GCF)<\/strong><\/h4>\n\n\n\n

The terms in the expression 2×3+4×2\u2212x2x^3 + 4x^2 – x all have a common factor of x<\/strong>. So, we factor that out first: 2×3+4×2\u2212x=x(2×2+4x\u22121)2x^3 + 4x^2 – x = x(2x^2 + 4x – 1)<\/p>\n\n\n\n

Now we are left with factoring the quadratic trinomial inside the parentheses: 2×2+4x\u221212x^2 + 4x – 1<\/p>\n\n\n\n

Step 2: Factor the Quadratic<\/strong><\/h4>\n\n\n\n

To factor 2×2+4x\u221212x^2 + 4x – 1, we can use the method of splitting the middle term<\/strong> or trial and error<\/strong>.<\/p>\n\n\n\n

We look for two numbers that multiply to 2\u22c5(\u22121)=\u221222 \\cdot (-1) = -2, and add up to 4. These numbers are not obvious<\/strong>, so we proceed by factoring using the ac method<\/strong>: 2×2+4x\u22121=(2x\u22121)(x+1)2x^2 + 4x – 1 = (2x – 1)(x + 1)<\/p>\n\n\n\n

But this is incorrect<\/strong> because: (2x\u22121)(x+1)=2×2+2x\u2212x\u22121=2×2+x\u22121\u22602×2+4x\u22121(2x – 1)(x + 1) = 2x^2 + 2x – x – 1 = 2x^2 + x – 1 \\neq 2x^2 + 4x – 1<\/p>\n\n\n\n

So that\u2019s the wrong pair. Let’s factor it correctly using quadratic factoring<\/strong>.<\/p>\n\n\n\n

Try factoring 2×2+4x\u221212x^2 + 4x – 1 by splitting the middle term:
We look for two numbers that multiply to 2\u22c5(\u22121)=\u221222 \\cdot (-1) = -2 and add to 44. No integers work, so instead, we use factoring by grouping<\/strong>:<\/p>\n\n\n\n

Multiply leading coefficient and constant:
a\u22c5c=2\u22c5(\u22121)=\u22122a \\cdot c = 2 \\cdot (-1) = -2<\/p>\n\n\n\n

We need to find two numbers that multiply to -2 and add up to 4. This is not possible with integers. So we factor the quadratic directly:<\/p>\n\n\n\n

Use the quadratic formula<\/strong>: x=\u22124\u00b1(4)2\u22124(2)(\u22121)2(2)=\u22124\u00b116+84=\u22124\u00b1244x = \\frac{-4 \\pm \\sqrt{(4)^2 – 4(2)(-1)}}{2(2)} = \\frac{-4 \\pm \\sqrt{16 + 8}}{4} = \\frac{-4 \\pm \\sqrt{24}}{4} =\u22124\u00b1264=\u22122\u00b162= \\frac{-4 \\pm 2\\sqrt{6}}{4} = \\frac{-2 \\pm \\sqrt{6}}{2}<\/p>\n\n\n\n

So the quadratic factors are: 2×2+4x\u22121=2(x\u2212\u22122+62)(x\u2212\u22122\u221262)2x^2 + 4x – 1 = 2\\left(x – \\frac{-2 + \\sqrt{6}}{2}\\right)\\left(x – \\frac{-2 – \\sqrt{6}}{2}\\right)<\/p>\n\n\n\n

But these are irrational. Let’s try factoring the original expression<\/strong> by grouping.<\/p>\n\n\n\n

Back to: 2×3+4×2\u2212x=x(2×2+4x\u22121)2x^3 + 4x^2 – x = x(2x^2 + 4x – 1)<\/p>\n\n\n\n

Try trial factoring<\/strong> again: x(2×2+4x\u22121)=x(2x\u22121)(x+1)x(2x^2 + 4x – 1) = x(2x – 1)(x + 1)<\/p>\n\n\n\n

Check: (2x\u22121)(x+1)=2×2+2x\u2212x\u22121=2×2+x\u22121(2x – 1)(x + 1) = 2x^2 + 2x – x – 1 = 2x^2 + x – 1<\/p>\n\n\n\n

Still incorrect. Let\u2019s go back.<\/p>\n\n\n\n

Try factoring the original polynomial directly by grouping: 2×3+4×2\u2212x=(2×3+4×2)\u2212x=2×2(x+2)\u22121(x+2)2x^3 + 4x^2 – x = (2x^3 + 4x^2) – x = 2x^2(x + 2) -1(x + 2) =(2×2\u22121)(x+2)= (2x^2 – 1)(x + 2)<\/p>\n\n\n\n

But this gives: (2×2\u22121)(x+2)=2×3+4×2\u2212x\u22122\u22602×3+4×2\u2212x(2x^2 – 1)(x + 2) = 2x^3 + 4x^2 – x – 2 \\neq 2x^3 + 4x^2 – x<\/p>\n\n\n\n

Eventually, the correct factorization (using rational root theorem and factoring) is: x(2x+1)(x\u22121)x(2x + 1)(x – 1)<\/p>\n\n\n\n

\u2705 Let’s check: (2x+1)(x\u22121)=2×2\u22122x+x\u22121=2×2\u2212x\u22121(2x + 1)(x – 1) = 2x^2 – 2x + x – 1 = 2x^2 – x – 1<\/p>\n\n\n\n

Then: x(2×2\u2212x\u22121)=2×3\u2212x2\u2212x\u2190 Incorrectx(2x^2 – x – 1) = 2x^3 – x^2 – x \\quad \\text{\u2190 Incorrect}<\/p>\n\n\n\n

So this is wrong.<\/p>\n\n\n\n

Let\u2019s go back to square one.<\/p>\n\n\n\n

Actually, best to factor by grouping<\/strong> the original polynomial:<\/p>\n\n\n\n

Group as: (2×3\u2212x)+4×2=x(2×2\u22121)+4×2(2x^3 – x) + 4x^2 = x(2x^2 – 1) + 4x^2<\/p>\n\n\n\n

This does not help.<\/p>\n\n\n\n

Now group differently: 2×3+4×2\u2212x=x(2×2+4x\u22121)2x^3 + 4x^2 – x = x(2x^2 + 4x – 1)<\/p>\n\n\n\n

Try factoring with the quadratic formula<\/strong> as above.<\/p>\n\n\n\n

So the factored form is: x(2×2+4x\u22121)orx(2x\u2212(\u22122+6))(x\u2212(\u22122\u22126))x\\left(2x^2 + 4x – 1\\right) \\quad \\text{or} \\quad x\\left(2x – (-2 + \\sqrt{6})\\right)\\left(x – (-2 – \\sqrt{6})\\right)<\/p>\n\n\n\n

That is: x(x+\u22122+62)(x+\u22122\u221262)x\\left(x + \\frac{-2 + \\sqrt{6}}{2}\\right)\\left(x + \\frac{-2 – \\sqrt{6}}{2}\\right)<\/p>\n\n\n\n

So the final correct factored form<\/strong> is: x(2×2+4x\u22121)\\boxed{x(2x^2 + 4x – 1)}<\/p>\n\n\n\n

And since this quadratic does not factor nicely over the rationals, this is the correct simplest rational factorization<\/strong>.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

What is the factored form of 2×3 + 4×2 \u2013 x? The Correct Answer and Explanation is: The factored form of the expression 2×3+4×2\u2212x2x^3 + 4x^2 – x is: x(2x+1)(x\u22121)x(2x + 1)(x – 1) Step-by-Step Explanation To factor the polynomial 2×3+4×2\u2212x2x^3 + 4x^2 – x, we follow a systematic approach using factoring techniques such as […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-225795","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225795","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=225795"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225795\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}