Examkin

Question

Difficulty: MediumPolynomial Factors and Graphs

For the polynomial p(x)=2x3−x2−kx+6p(x) = 2x^3 - x^2 - kx + 6, where kk is a constant, the remainder when p(x)p(x) is divided by 2x−32x - 3 is 00. What is the value of kk?

Answer: 7

Answer

The value of kk is 77.
The correct answer is 77. According to the Factor Theorem, a linear expression ax−bax - b is a factor of a polynomial p(x)p(x) if and only if p(ba)=0p\left(\frac{b}{a}\right) = 0. Here, the divisor is 2x−32x - 3, so setting 2x−3=02x - 3 = 0 gives the root x=32x = \frac{3}{2}. Substituting x=32x = \frac{3}{2} into p(x)=2x3−x2−kx+6p(x) = 2x^3 - x^2 - kx + 6 gives 2(32)3−(32)2−k(32)+6=02\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0. Simplifying this equation gives 274−94−32k+6=0\frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0, which reduces to 92−32k+6=0\frac{9}{2} - \frac{3}{2}k + 6 = 0. Combining the constant terms gives 212−32k=0\frac{21}{2} - \frac{3}{2}k = 0, which simplifies to 3k=213k = 21. Solving for kk gives k=7k = 7.

Step-by-Step Solution

1
Determine the root associated with the linear factor 2x−32x - 3.
Setting 2x−3=02x - 3 = 0 gives x=32x = \frac{3}{2}. By the Factor Theorem, p(32)=0p\left(\frac{3}{2}\right) = 0.
According to the Factor Theorem, a polynomial p(x)p(x) has a factor of the form ax−bax - b if and only if p(ba)=0p\left(\frac{b}{a}\right) = 0.
2
Substitute x=32x = \frac{3}{2} into the polynomial expression 2x3−x2−kx+62x^3 - x^2 - kx + 6 and set it to 00.
2(32)3−(32)2−k(32)+6=02\left(\frac{3}{2}\right)^3 - \left(\frac{3}{2}\right)^2 - k\left(\frac{3}{2}\right) + 6 = 0
This establishes a linear equation in terms of the unknown constant kk.
3
Simplify the numerical terms and solve for kk.
2(278)−94−32k+6=0  ⟹  274−94−32k+6=0  ⟹  92−32k+6=0  ⟹  212−32k=0  ⟹  3k=21  ⟹  k=72\left(\frac{27}{8}\right) - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{27}{4} - \frac{9}{4} - \frac{3}{2}k + 6 = 0 \implies \frac{9}{2} - \frac{3}{2}k + 6 = 0 \implies \frac{21}{2} - \frac{3}{2}k = 0 \implies 3k = 21 \implies k = 7.
Standard fractional arithmetic and algebraic isolation are used to find the value of kk.

Key Concept

Factor and Remainder Theorems
Rate this question

Topics

Polynomial Factors and Graphs

More questions from this topic