Question

5. If p(2) = -3 and p(3) = 17 in the polynomial
p(x) = 3x3 - 7x2 + ax + b, find the values of a
and b.​ please help me guys

Answer 1

Given:-

• $p(x) = 3 {x}^{3} - 7 {x}^{2} + ax + b$
• $p(2) = - 3$
• $p(3) = 17$

To find:-

• The value of a and b.

Answer:-

If p(x) is a polynomial, and we want to find p(a), we just have to substitute x = a.

Given that,

$p(x) = 3 {x}^{3} - 7 {x}^{2} + ax + b$

p(2):

$p(2) = 3( {2}^{3} ) - 7( {2}^{2} ) + 2a + b$

According to the question, p(2) = -3. So,

$\longrightarrow 3( {2}^{3} ) - 7( {2}^{2} ) + 2a + b = - 3$

$\longrightarrow 3( 8 ) - 7( 4 ) + 2a + b = - 3$

$\longrightarrow 24 - 28+ 2a + b = - 3$

$\longrightarrow - 4+ 2a + b = - 3$

$\longrightarrow 2a + b = 1 \quad \quad \dots(1)$

p(3):

$p(3) = 3( {3}^{3} ) - 7( {3}^{2} ) + 3a + b$

According to the question, p(3) = 17. So,

$\longrightarrow 3( {3}^{3} ) - 7( {3}^{2} ) + 3a + b = 17$

$\longrightarrow 3( 27 ) - 7( 9 ) + 3a + b = 17$

$\longrightarrow 81- 63 + 3a + b = 17$

$\longrightarrow 18 + 3a + b = 17$

$\longrightarrow 3a + b = - 1 \quad \quad \dots(2)$

Subtracting (1) from (2),

$3a - 2a = -1 - 1$

$\longrightarrow a = -2$

Putting this value of a in (1),

$2(-2) + b = 1$

$\longrightarrow b = 5$

$\longrightarrow \underline{ \underline{a = -2 \: and \: b = 5}}$