Question

for the polynomial function ( a(x) = x^4 - 2x^3 - 21x^2 + 22x + 40 ) we know ( (x - 5) ) is a factor. select all the other linear factors of ( a(x) ).
a ( square (x + 1) )
b ( square (x - 1) )
c ( square (x + 2) )
d ( square (x - 2) )
e ( square (x + 4) )
f ( square (x - 4) )
g ( square (x + 8) )

Explanation:

Step1: Divide the polynomial by \((x - 5)\)

We use polynomial long division or synthetic division. Let's use synthetic division for \(A(x)=x^{4}-2x^{3}-21x^{2}+22x + 40\) and root \(x = 5\).
The coefficients are \(1,-2,-21,22,40\).
Bring down the \(1\). Multiply by \(5\): \(1\times5 = 5\). Add to \(-2\): \(-2 + 5=3\).
Multiply \(3\) by \(5\): \(3\times5 = 15\). Add to \(-21\): \(-21+15=-6\).
Multiply \(-6\) by \(5\): \(-6\times5=-30\). Add to \(22\): \(22-30=-8\).
Multiply \(-8\) by \(5\): \(-8\times5=-40\). Add to \(40\): \(40 - 40=0\).
So the quotient polynomial is \(x^{3}+3x^{2}-6x - 8\).

Step2: Factor the cubic polynomial

Try to find a root of \(x^{3}+3x^{2}-6x - 8\). Test \(x = 2\): \(2^{3}+3\times2^{2}-6\times2 - 8=8 + 12-12 - 8 = 0\). So \((x - 2)\) is a factor.
Divide \(x^{3}+3x^{2}-6x - 8\) by \((x - 2)\) using synthetic division. Coefficients: \(1,3,-6,-8\), root \(x = 2\).
Bring down \(1\). Multiply by \(2\): \(2\). Add to \(3\): \(5\).
Multiply \(5\) by \(2\): \(10\). Add to \(-6\): \(4\).
Multiply \(4\) by \(2\): \(8\). Add to \(-8\): \(0\).
Quotient is \(x^{2}+5x + 4\).

Step3: Factor the quadratic polynomial

Factor \(x^{2}+5x + 4=(x + 1)(x + 4)\).

So the factors of \(A(x)\) are \((x - 5)\), \((x - 2)\), \((x + 1)\), \((x + 4)\). Comparing with the options:

• A. \((x + 1)\) is a factor.
• D. \((x - 2)\) is a factor.
• E. \((x + 4)\) is a factor.

Answer:

A. \((x + 1)\), D. \((x - 2)\), E. \((x + 4)\)