Question

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

Answer 1

`\huge \boxed{\mathbb{QUESTION} \downarrow}`

`\tt \: 2 { x }^( 4 ) + { x }^( 3 ) -29 { x }^( 2 ) -34x+24`

`\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}`

`\tt2 { x }^( 4 ) + { x }^( 3 ) -29 { x }^( 2 ) -34x+24`

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.

`\tt\left(x-4\right)\left(2x^(3)+9x^(2)+7x-6\right)`

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.

`\tt \: \left(x+3\right)\left(2x^(2)+3x-2\right)`

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.

`\tt \: a+b=3 \\ \tt \: ab=2\left(-2\right)=-4`

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.

`\tt-1,4 \\ \tt-2,2`

Calculate the sum for each pair.

`\tt-1+4=3 \\ \tt-2+2=0`

The solution is the pair that gives sum 3.

`\tt \: a=-1 \: \\ \tt b=4`

Rewrite
`\tt2x^(2)+3x-2` as
`\tt\left(2x^(2)-x\right)+\left(4x-2\right)`.

`\tt\left(2x^(2)-x\right)+\left(4x-2\right)`

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

`\tt \: x\left(2x-1\right)+2\left(2x-1\right)`

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

`\tt\left(2x-1\right)\left(x+2\right)`

Rewrite the complete factored expression.

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