Question

Is x -1 a factor of the polynomial 12x⁴ - 5x³ + 3x² - 5? Explain your answer.

Answer 1

I'd be more than glad to help you! <3

Full Explanation:

To determine if \( x - 1 \) is a factor of the polynomial \( 12x^{4} + 5x^{3} + 3x^{2} - 5 \), we can use the Remainder Theorem.

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), then the remainder is equal to \( f(a) \).

So, let's substitute \( x = 1 \) into the polynomial and see if the result is zero.

\( f(1) = 12(1)^{4} + 5(1)^{3} + 3(1)^{2} - 5 \)

Simplifying, we get:

\( f(1) = 12 + 5 + 3 - 5 \)

\( f(1) = 15 \)

Answer 2

To check if x - 1 is a factor of the polynomial, we apply the Remainder Theorem. Substituting x = 1 into the polynomial yields a remainder of zero, confirming that x - 1 is a factor.

Step-by-step explanation:

The student has asked if x - 1 is a factor of the polynomial 12x⁴ - 5x³ + 3x² - 5. To determine if x - 1 is indeed a factor, we must perform polynomial division or apply the Remainder Theorem. The Remainder Theorem states that for a polynomial p(x), the remainder of the division of p(x) by x - a is p(a). Therefore, if x - 1 is a factor of 12x⁴ - 5x³ + 3x² - 5, then the polynomial will yield a remainder of zero when we evaluate it at x = 1.

To apply the Remainder Theorem, we substitute x = 1 into the polynomial and evaluate:

• 12(1)⁴ - 5(1)³ + 3(1)² - 5
• 12 - 5 + 3 - 5
• 10 - 10
• 0

Since the result is zero, it confirms that x - 1 is indeed a factor of the given polynomial.