Polynomial function h is defined as h(x)=2x^(3)-9x^(2)+cx-6, where c is a constant. If 2x-3 is a factor of the polynomial, then what is the value of c ? ◻

Question

Polynomial function h is defined as h(x)=2x^(3)-9x^(2)+cx-6, where c is a constant. If  2x-3 is a factor of the polynomial, then what is the value of c ? ◻

Bytelearn · Accepted Answer

Apply Factor Theorem: Since 2x - 3 is a factor of the polynomial h(x), we can use the Factor Theorem which states that if (ax - b) is a factor of a polynomial, then the polynomial will equal zero when x equals b/a.Find x value: Let's find the value of x that makes 2x - 3 equal to zero. We solve the equation 2x - 3 = 0 for x. 2x = 3 x = 3/2Substitute x into h(x): Now we substitute x = 3/2 into the polynomial h(x) and set it equal to zero, because the Factor Theorem tells us that h(3/2) should be zero if 2x - 3 is indeed a factor. h(3/2) = 2(3/2)^3 - 9(3/2)^2 + c(3/2) - 6 = 0Calculate h(3/2): We calculate each term separately: (3/2)^3 = 27/8 (3/2)^2 = 9/4 Now we substitute these values into the equation: 2(27/8) - 9(9/4) + c(3/2) - 6 = 0Simplify terms: Simplify each term: 2(27/8) = 27/4 9(9/4) = 81/4 c(3/2) = 3c/2 Now we have: 27/4 - 81/4 + 3c/2 - 6 = 0Combine like terms: Combine like terms and simplify the equation: (27/4 - 81/4) + 3c/2 - 6 = 0 (-54/4) + 3c/2 - 6 = 0 -13.5 + 3c/2 - 6 = 0Combine constant terms: Combine the constant terms: -13.5 - 6 = -19.5 So we have: -19.5 + 3c/2 = 0Solve for c: Now we solve for c: 3c/2 = 19.5 Multiply both sides by 2/3 to isolate c: c = (19.5) * (2/3) c = 39/3 c = 13

Polynomial function $h$ is defined as $h(x)=2x^{3}-9x^{2}+cx-6$ , where $c$ is a constant. If $2x-3$ is a factor of the polynomial, then what is the value of $c$ ? $\newline$ $\square$

Full solution

More problems from Simplify exponential expressions using exponent rules

Related topics

Polynomial function hhh is defined as h(x)=2x3−9x2+cx−6h(x)=2x^{3}-9x^{2}+cx-6h(x)=2x3−9x2+cx−6, where ccc is a constant. If 2x−32x-32x−3 is a factor of the polynomial, then what is the value of ccc ?\newline□\square□

Polynomial function $h$ is defined as $h(x)=2x^{3}-9x^{2}+cx-6$ , where $c$ is a constant. If $2x-3$ is a factor of the polynomial, then what is the value of $c$ ? $\newline$ $\square$