Which polynomial is factored completely
a. 4 (4x^4-1)
b. 2x (y^3-4y^2+5y)
c. 3x (9\u00d7^2+1)
d. 5x^2-17x+14?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The correct answer is: d. ( 5x^2 – 17x + 14 )<\/strong><\/p>\n\n\n\n To determine which polynomial is factored completely, let’s analyze each option one by one:<\/p>\n\n\n\n This expression can be factored further. The term ( 4x^4 – 1 ) is a difference of squares and can be factored as: First, factor out the greatest common factor (GCF) in the polynomial: The term ( 9x^2 + 1 ) does not factor further over the reals because it is a sum of squares. Therefore, the factorization remains: To factor this quadratic polynomial, we need to find two numbers that multiply to ( 5 \\times 14 = 70 ) and add to (-17). The numbers (-5) and (-14) satisfy these conditions. Therefore, we can factor it as: The only polynomial that is factored completely is d. ( 5x^2 – 17x + 14 )<\/strong>, which factors to ( (5x – 14)(x – 1) ). Options a, b, and c can all be factored further, indicating that they are not in their completely factored forms.<\/p>\n","protected":false},"excerpt":{"rendered":" Which polynomial is factored completelya. 4 (4x^4-1)b. 2x (y^3-4y^2+5y)c. 3x (9\u00d7^2+1)d. 5x^2-17x+14? The Correct Answer and Explanation is : The correct answer is: d. ( 5x^2 – 17x + 14 ) To determine which polynomial is factored completely, let’s analyze each option one by one: a. ( 4(4x^4 – 1) ) This expression can be […]<\/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-152061","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/152061","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=152061"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/152061\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=152061"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=152061"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=152061"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}a. ( 4(4x^4 – 1) )<\/h3>\n\n\n\n
[
4x^4 – 1 = (2x^2 – 1)(2x^2 + 1)
]
Thus, the complete factorization is:
[
4(2x^2 – 1)(2x^2 + 1)
]
This means option (a) is not fully factored.<\/p>\n\n\n\nb. ( 2x(y^3 – 4y^2 + 5y) )<\/h3>\n\n\n\n
[
y^3 – 4y^2 + 5y = y(y^2 – 4y + 5)
]
Now, we need to factor ( y^2 – 4y + 5 ). The discriminant is ( (-4)^2 – 4(1)(5) = 16 – 20 = -4 ), which is negative, indicating no real roots. Therefore, ( y^2 – 4y + 5 ) cannot be factored further with real numbers. Thus, the factorization is:
[
2xy(y^2 – 4y + 5)
]
This indicates option (b) is not completely factored.<\/p>\n\n\n\nc. ( 3x(9x^2 + 1) )<\/h3>\n\n\n\n
[
3x(9x^2 + 1)
]
This means option (c) is not completely factored.<\/p>\n\n\n\nd. ( 5x^2 – 17x + 14 )<\/h3>\n\n\n\n
[
5x^2 – 5x – 14x + 14 = 5x(x – 1) – 14(x – 1) = (5x – 14)(x – 1)
]
Thus, option (d) is fully factored.<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n