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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?

Cevap: 7

Cevap

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.

Adım Adım Çözüm

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.

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Factor and Remainder Theorems
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