{"id":5122,"date":"2021-12-02T19:48:38","date_gmt":"2021-12-02T19:48:38","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=5122"},"modified":"2021-12-21T11:39:16","modified_gmt":"2021-12-21T11:39:16","slug":"factor-theorem-with-examples-and-faqs","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/factor-theorem-with-examples-and-faqs\/","title":{"rendered":"Factor Theorem with Examples and FAQs"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Factor Theorem with Examples and FAQs<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

Factor Theorem<\/b><\/h2>\n

The factor theorem is used for factorizing a polynomial, which helps to determine the roots of the polynomial. Factor theorem is known to be a special case of a polynomial remainder theorem.<\/span><\/p>\n

 <\/p>\n

Statement:<\/b><\/h2>\n

let f(x)<\/span> be a polynomial of\u00a0 degree n>1 and a be any real number<\/p>\n\\text { (1) If } f(a)=0 \\text { then }(x-a) \\text { is a factor of } f(x) \\text {. }<\/span>\n\\text { (2) If }(x-a) \\text { is a factor of } f(x) \\text { then } f(a)=0 \\text {. }<\/span>\n

 <\/p>\n

Proof:<\/b><\/h2>\n

\\text { (1) Let } f(a)=0 \\text {. }<\/span><\/b><\/p>\n

On the division of f(x) \\text { by }(x-a)<\/span><\/span>, <\/span>we will get a quotient.<\/span><\/p>\n

Let the quotient be q(x)<\/span><\/span>.<\/span><\/p>\n

By The Remainder Theorem, when <\/span>f<\/span>x<\/span> is divided by <\/span>x-a<\/span>, then the remainder is\u00a0 f(a)<\/span><\/span>.<\/span><\/p>\n

\\therefore f(x)=(x-a) \\cdot q(x)+f(a)<\/span><\/span><\/p>\n\\Rightarrow f(x)=(x-a) \\cdot q(x), \\quad[\\because f(a)=0]<\/span>\n\\Rightarrow(x-a) \\text { is a factor of } f(x)<\/span>\n

 <\/p>\n

(2) Let <\/span>x-a<\/span> is a factor of f(x)<\/span><\/span>.<\/span><\/p>\n

On the division of f(x)<\/span><\/span>by (<\/span>x-a)<\/span>, <\/span>we will get a quotient. Let the quotient be q(x)<\/span><\/span>.<\/span><\/p>\n

\\therefore f(x)=(x-a) \\cdot q(x)<\/span><\/span><\/p>\n

Putting x = a\u00a0 in the above equation we get,\u00a0<\/span><\/p>\n

f(a)=(a-a) \\cdot q(a) \\Rightarrow f(a)=0 \\cdot q(x)<\/span><\/span><\/p>\n\\therefore f(a)=0<\/span>\n

Thus, <\/span>x-a<\/span> is a factor of f(x) \\Rightarrow f(a)=0<\/span><\/span><\/p>\n

For example, Consider the polynomial function f(x)=x^{2}-17 x+16<\/span><\/span><\/p>\n

Let\u2019s find the values of <\/span>x<\/span> for which f(x)=0<\/span><\/span>.<\/span><\/p>\n

Solve the equation, assuming f(x)=0<\/span><\/span><\/p>\n

\\Rightarrow x^{2}-17 x+16=0 <\/span><\/span><\/p>\n\\Rightarrow x^{2}-16 x-x+16=0 <\/span>\n\\Rightarrow(x-16)(x-1)=0 <\/span>\nx=16 \\text { or } x=1<\/span>\n

 <\/p>\n

\u2234x-16 and x-1 are factors of fx and x=16 or\u00a0 x=1 are the solution to the equation x^{2}-17 x+16=0<\/span><\/span><\/p>\n

Now to do the converse, and check if x-16 and x-1 are factors of fx, we will put the value of x as x=16 and\u00a0 x=1, one by one in the polynomial function, f(x)=x^{2}-17 x+16<\/span><\/span><\/span><\/p>\n

For x=16,<\/span><\/p>\n

f(16)=16^{2}-17(16)+16=256-272+16=0<\/span><\/span><\/p>\n

\\Rightarrow f(16)=0<\/span><\/span><\/p>\n

For <\/span>x=1<\/span>,<\/span><\/p>\n

f(1)=1^{2}-17(1)+16=1-17+16=0<\/span><\/span><\/p>\n

\\Rightarrow f(1)=0<\/span><\/span><\/p>\n

Thus, x-16 and x-1 are factors of f(x)<\/span>.<\/span><\/p>\n

<\/span><\/p>\n

Example<\/b><\/h2>\n

Use the factor theorem to check if (x-5) is a factor of f(x)=3 x^{2}-8 x-35<\/span><\/span><\/b><\/p>\n

Solution<\/b><\/p>\n

x-5=0\u21d2x=5<\/span><\/p>\n

By factor theorem, (<\/span>x-5)<\/span>\u00a0will be a factor of <\/span>f<\/span>x<\/span>=3<\/span>x<\/span>2<\/span>-8x-35<\/span>, if f(x)=3 x^{2}-8 x-35, \\text { if } f(5)=0<\/span><\/span>.<\/span><\/p>\n

f(x)=3 x^{2}-8 x-35<\/span><\/span><\/p>\n

f(5)=3(5)^{2}-8(5)-35=75-40-35=0 <\/span><\/span><\/p>\n

\\Rightarrow(x-5) \\text { is a factor of } f(x)=3 x^{2}-8 x-35<\/span>
<\/span><\/p>\n

Hence<\/span> (x-5)<\/span>\u00a0is a factor of the given polynomial f(x)<\/span><\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

 <\/p>\n

 <\/p>\n

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Explore Other Topics<\/h1>\n

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[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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\n
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Reflex angle<\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1: What is the factor theorem?
    <\/strong><\/h3>\n

    Ans: <\/strong>The statement of factor theorem is :<\/span><\/span><\/p>\n

    let f(x)<\/span>be a polynomial of degree n>1 and let a be any real number.<\/span><\/span><\/p>\n

    (1) If f(a)=0\u00a0 then (x-a)\u00a0 is\u00a0 \u00a0a\u00a0 factor\u00a0 of\u00a0 \u00a0 f(x).<\/span><\/p>\n

    (2) If\u00a0 \u00a0(x-a)\u00a0 \u00a0is\u00a0 a\u00a0 factor\u00a0 of\u00a0 f (x) then\u00a0 f(a)=0.<\/span><\/span><\/p>\n

    <\/span><\/p>\n

    <\/span><\/p>\n

    Q2. What is the significance of using the factor theorem?
    <\/strong><\/h3>\n

    Ans: <\/strong>The factor theorem is used for factoring a polynomial, which helps to determine the roots or zeros of the polynomial.
    <\/strong><\/p>\n

     <\/p>\n

    Q3. How to find if ( x-a) is a factor of a polynomial f(x)<\/span>?<\/strong><\/h3>\n

    Ans: <\/strong>( x-a)<\/span> is a factor of a polynomial f(x)<\/span> <\/span>if and only if, forx=a, f(x)=0<\/span><\/span>.<\/span><\/p>\n

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