Question

which is not a factor of $x^4 - x^3 - 12x^2 - 4x + 16$? (hint: find the remainders) options: $x - 4$, $x + 2$, $x - 1$, $x + 1$

Explanation:

Step1: Define the polynomial

Let \( f(x) = x^4 - x^3 - 12x^2 - 4x + 16 \)

Step2: Test \( x - 4 \) (root \( x=4 \))

\( f(4) = 4^4 - 4^3 -12(4)^2 -4(4)+16 = 256 -64 -192 -16 +16 = 0 \). So \( x-4 \) is a factor.

Step3: Test \( x + 2 \) (root \( x=-2 \))

\( f(-2) = (-2)^4 - (-2)^3 -12(-2)^2 -4(-2)+16 =16 +8 -48 +8 +16=0 \). So \( x+2 \) is a factor.

Step4: Test \( x -1 \) (root \( x=1 \))

\( f(1)=1^4 -1^3 -12(1)^2 -4(1)+16=1 -1 -12 -4 +16=0? \) Wait correction: \(1 -1 -12 -4 +16=0\)? No: \(1-1=0;0-12=-12;-12-4=-16;-16+16=0\)? Wait no wait calculation again: \(1 -1 -12 -4 +16 = (1-1)+(-12-4)+16=0-16+16=0\)? Wait wait another way: factor the polynomial. Let's divide \( f(x) \) by \( (x-4)(x+2)=x²-2x-8 \). Using polynomial division: \(x⁴ -x³ -12x² -4x +16 ÷ (x²-2x-8)\). First term: \(x²\), multiply divisor by \(x²\): \(x⁴-2x³-8x²\). Subtract from dividend: \( (x⁴ -x³ -12x²) - (x⁴-2x³-8x²)=x³ -4x² \). Bring down next terms: \(x³ -4x² -4x +16\). Next term: \(x\), multiply divisor by \(x\): \(x³-2x²-8x\). Subtract: \( (x³-4x²-4x) - (x³-2x²-8x)= -2x² +4x \). Bring down 16: \( -2x² +4x +16 \). Next term: -2, multiply divisor by -2: \( -2x² +4x +16 \). Subtract: 0. So quotient is \(x² +x -2=(x+2)(x-1)\). Wait so factors are \( (x-4)(x+2)(x+2)(x-1) \). Wait then why? Wait wait no wait original polynomial: \(x⁴ -x³ -12x² -4x +16=(x-4)(x+2)^2(x-1)\). Then all options are factors? Wait no wait the image options: wait the user's image has options \(x-4\), \(x+2\), \(x-1\), \(x+1\). Oh right! I missed \(x+1\). Test \(x+1\) (root \(x=-1\)): \(f(-1)=(-1)^4 - (-1)^3 -12(-1)^2 -4(-1)+16=1 +1 -12 +4 +16=10≠0\). So the non-factor is \(x+1\)? Wait wait the user's image: let's recheck. The user's image shows options: top is \(x-4\), then \(x+2\), then \(x-1\), then \(x+1\). The question is which is NOT a factor. So \(x+1\) is not a factor.

Step5: Correct test for \(x+1\)

\(f(-1)=1 +1 -12 +4 +16=10≠0\), so \(x+1\) is not a factor.

Final Answer:

x + 1

Answer:

x - 1