If (2x - 3) is a factor of the polynomial P(x) = 2x^3 - 5x^2 + ax + 6, what is the value o… — JAMB UTME

If $(2x - 3)$ is a factor of the polynomial $P(x) = 2x^3 - 5x^2 + ax + 6$ , what is the value of the constant $a$ ?

The value of the constant

a

-1

According to the Factor Theorem, if

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

. Substituting

x = \frac{3}{2}

gives

\frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

, which simplifies to

\frac{3}{2} + \frac{3}{2}a = 0

, resulting in

a = -1

Apply the Factor Theorem by setting the linear factor equal to zero.

2x - 3 = 0 \implies x = \frac{3}{2}

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

Substitute

x = \frac{3}{2}

into the polynomial expression

P(x)

P\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 - 5\left(\frac{3}{2}\right)^2 + a\left(\frac{3}{2}\right) + 6 = 0

Setting the resulting expression equal to zero allows solving for

a

Simplify the powers and numerical terms.

2\left(\frac{27}{8}\right) - 5\left(\frac{9}{4}\right) + \frac{3}{2}a + 6 = 0 \implies \frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

Evaluate each fraction before combining terms.

Combine constant terms and solve for

a

-\frac{18}{4} + 6 + \frac{3}{2}a = 0 \implies -\frac{9}{2} + 6 + \frac{3}{2}a = 0 \implies \frac{3}{2} + \frac{3}{2}a = 0 \implies a = -1

Isolating

a

yields the correct constant value.

Factor Theorem for linear divisors of the form

(ax - b)

Soru