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Which expressions are factors of this polynomial? 2x^4 + x^3 - 29x^2 - 34x + 24

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[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]

[tex] \tt \: 2 { x }^{ 4 } + { x }^{ 3 } -29 { x }^{ 2 } -34x+24[/tex]

[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]

[tex] \tt2 { x }^{ 4 } + { x }^{ 3 } -29 { x }^{ 2 } -34x+24[/tex]

By Rational Root Theorem, all rational roots of a polynomial are in the form p/q, where p divides the constant term 24 and q divides the leading coefficient 2. One such root is 4. Factor the polynomial by dividing it by x-4.

[tex] \tt\left(x-4\right)\left(2x^{3}+9x^{2}+7x-6\right) [/tex]

Consider 2x³+9x²+7x-6. By Rational Root Theorem, all rational roots of a polynomial are in the form p/q, where p divides the constant term -6 and q divides the leading coefficient 2. One such root is -3. Factor the polynomial by dividing it by x+3.

[tex] \tt \: \left(x+3\right)\left(2x^{2}+3x-2\right) [/tex]

Consider 2x²+3x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 2x²+ax+bx-2. To find a and b, set up a system to be solved.

[tex] \tt \: a+b=3 \\ \tt \: ab=2\left(-2\right)=-4 [/tex]

As ab is negative, a and b have the opposite signs. As a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4.

[tex] \tt-1,4 \\ \tt-2,2 [/tex]

Calculate the sum for each pair.

[tex] \tt-1+4=3 \\ \tt-2+2=0 [/tex]

The solution is the pair that gives sum 3.

[tex] \tt \: a=-1 \: \\ \tt b=4 [/tex]

Rewrite [tex]\tt2x^{2}+3x-2[/tex] as [tex]\tt\left(2x^{2}-x\right)+\left(4x-2\right)[/tex].

[tex] \tt\left(2x^{2}-x\right)+\left(4x-2\right) [/tex]

Exclude x in the first group and 2 in the second group.

[tex] \tt \: x\left(2x-1\right)+2\left(2x-1\right) [/tex]

Factor out common term 2x-1 by using distributive property.

[tex] \tt\left(2x-1\right)\left(x+2\right) [/tex]

Rewrite the complete factored expression.

[tex] \underline{ \tt \: Factors \rightarrow} \boxed{\boxed{ \bf\left(x-4\right)\left(2x-1\right)\left(x+2\right)\left(x+3\right) }}[/tex]

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