Question

28.Verify that whether -1 is zero of the polynomial p(x) = 2x

3

- 9x

2 + x + 12 and find the

value of p(0), p(1), p(2).​

Answer 1

Correct Question

• Verify that whether -1 is a zero of the polynomial p(x) = 2x³ - 9x² + x + 12 and find the value of p(0), p(1) and p(2).

⠀

Given Equation

• p(x) = 2x³ - 9x² + x + 12.

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To prove

• -1 is zero of the given polynomial.

⠀

To find

• Value of

⠀⠀⠀⠀⠀⠀⠀⠀→ p(0)

⠀⠀⠀⠀⠀⠀⠀⠀→ p(1)

⠀⠀⠀⠀⠀⠀⠀⠀→ p(2)

⠀

Solution

• Firstly, we will verify whether -1 is a zero of the given polynomial.

⠀

$\tt\longrightarrow{p(x) = 2x^3 - 9x^2 + x + 12}$

⠀

$\tt\longrightarrow{p(-1) = 2(-1)^3 - 9(-1)^2 + (-1) + 12}$

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$\tt\longrightarrow{p(-1) = 2(-1) - 9(1) - 1 + 12}$

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$\tt\longrightarrow{p(-1) = -2 - 9 + 11}$

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$\tt\longrightarrow{p(-1) = -11 + 11}$

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$\tt\longrightarrow{p(-1) = 0}$

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★ We find that p(-1) = 0

• Hence it is proved that -1 is a zero of given polynomial.

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$\: \: \: \: \: \: \: \underline{\sf{\red{Finding\: the\: values}}}$

⠀

$\large{\boxed{\boxed{\sf{Case\: I}}}}$

★ x = 0

⠀

$\tt:\implies{p(0) = 2(0)^3 - 9(0)^2 + (0) + 11}$

$\tt:\implies{p(0) = 2(0) - 9(0) + 0 + 11}$

$\tt:\implies{p(0) = 0 - 0 + 11}$

$\bf:\implies{p(0) = 11}$

• p(0) = 11

⠀

$\large{\boxed{\boxed{\sf{Case\: II}}}}$

★ x = 1

⠀

$\tt:\implies{p(1) = 2(1)^3 - 9(1)^2 + (1) + 11}$

$\tt:\implies{p(1) = 2(1) - 9(1) + 1 + 11}$

$\tt:\implies{p(1) = 2 - 9 + 12}$

$\bf:\implies{p(1) = 5}$

• p(1) = 5

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$\large{\boxed{\boxed{\sf{Case\: III}}}}$

★ x = 2

⠀

$\tt:\implies{p(2) = 2(2)^3 - 9(2)^2 + (2) + 11}$

$\tt:\implies{p(2) = 2(8) - 9(2) + 2 + 11}$

$\tt:\implies{p(2) = 16 - 18 + 13}$

$\bf:\implies{p(2) = 11}$

• p(2) = 11

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Hence,

• The values of

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀→ p(0) = 11

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀→ p(1) = 5

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀→ p(2) = 11

⠀

━━━━━━━━━━━━━━━━━━━━━━

Answer 2

Answer:

p(0) = 12

p(1) = 6

p(2) = 4

Step-by-step explanation:

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