Question

If (x-2) and (x+3) are factor are x³ + ax²+bx-30 find a nad b

Answer 1

Given:

(x – 2) and (x + 3) are the factors of x³ + ax² + bx – 30.

To find:

We've to find the values of 'a' and 'b'.

Required answer:

The value of 'a' and 'b' when (x – 2) and (x + 3) are the factors of the polynomial x³ + ax² + bx – 30 are 6 and – 1 respectively.

Explaination:

(x – 2) and (x + 3) are the factors of x³ + ax² + bx – 30.

This means that 2, – 3 when substituted in the given polynomial, result to zero.

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So, now let's substitute these values in the polynomial and get a linear equation :

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For x = 2:

$\sf : \implies \: \: {x}^{3} + a {x}^{2} + bx - 30 = 0$

$\sf : \implies \: \: {(2)}^{3} + a {(2)}^{2} + b(2) - 30 = 0$

$\sf : \implies \: \: 8 + 4a + 2b - 30 = 0$

$\sf : \implies \: \: 4a + 2b - 22 = 0 \: \: \: \{dividing \: the \: whole \: by \: 2 \}$

$\sf : \implies \: \: 2a + b = 11 \: \: \:$

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For x = – 3:

$\sf : \implies \: \: {x}^{3} + a {x}^{2} + bx + c = 0$

$\sf : \implies \: \: {( - 3)}^{3} + a( - 3) ^{2} + b( - 3) - 30 = 0$

$\sf : \implies \: \: - 27 + 9a - 3b - 30 = 0$

$\sf : \implies \: \: 9a - 3b - 57 = 0$

$\sf : \implies \: \: 9a - 3b = 57 \: \: \: \{dividing \: the \: whole \: by \: 3 \}$

$\sf : \implies \: \: 3a - b = 19 \:$

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The pair of linear equations in two variables are :

$\sf : \implies \: \: 2a + b = 11 \: \: \: \dots \dots \: {eq}^{n} \{i \}$

$\sf : \implies \: \: 3 a - b = 19 \: \: \: \dots \dots \: {eq}^{n } \{ii \}$

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Getting the value of 'b' from eqⁿ{ii} :

$\sf : \implies \: \: 3a - b = 19$

$\sf : \implies \: \: b = 3a - 19$

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Substituting this value of 'b' in eqⁿ{i} :

$\sf : \implies \: \: 2a + b = 19$

$\sf : \implies \: \: 2a + ( 3a - 19) = 11$

$\sf : \implies \: \: 2a + 3 a - 19 = 11$

$\sf : \implies \: \: 5 a = 19 + 11$

$\sf : \implies \: \: 5a = 30$

$\sf : \implies \: \: a = \cancel\dfrac{30}{5}$

$\sf : \implies \: \: \boxed{ \sf \red{a = 6}}$

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Substituting the 'a' value in eqⁿ{ii} :

$\sf : \implies \: \: 3a - b = 19$

$\sf : \implies \: \: 3(6) - b = 19$

$\sf : \implies \: \: 18 - b = 19$

$\sf : \implies \: \: - b = 19 - 18$

$\sf : \implies \: \: - b = 1$

$: \implies \: \: \boxed{ \sf \red{b = - 1}}$