Q. Polynomial function h is defined as h(x)=2x3−9x2+cx−6, where c is a constant. If 2x−3 is a factor of the polynomial, then what is the value of c ?
Factor Theorem Application: 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 ab. In this case, we can find the value of x that makes 2x−3 equal to zero.2x−3=02x=3x=23
Substitute x Value: Now we substitute x=23 into the polynomial h(x) and set it equal to zero, because 2x−3 is a factor of h(x).h(23)=2(23)3−9(23)2+c(23)−6=0
Calculate Polynomial: Let's calculate each term separately.First term: 2(23)3=2×827=854=427Second term: −9(23)2=−9×49=−481Third term: c(23)=23cFourth term: −6Now we combine these terms.427−481+23c−6=0
Combine Terms: Combine like terms and simplify the equation.(427−481)+23c−6=0(−454)+23c−6=0−227+23c−6=0
Combine Fractions: Now we need to combine all terms with fractions and the constant term.To combine −227 and −6, we need to express −6 as a fraction with a denominator of 2.−6=−212So the equation becomes:−227+23c−212=02−27−12+3c=02−39+3c=0
Solve for Constant c: To find the value of c, we need to solve for c in the equation 2−39+3c=0. Multiply both sides by 2 to get rid of the denominator.2×2−39+3c=2×0−39+3c=0Now we solve for c.3c=39c=339c=13
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