Question

pls help meeee very urgent​

Answer 1

Step-by-step explanation:

polynomial , x³ -ax² -13x + b has two factors ( x -1) and (x +3)

it means , x = 1 and -3 are the roots (zeros ) of x³ -ax² -13x + b .

so,

put x = 1 in polynomial ,.

(1)³ -a(1)² -13(1) + b = 0

1 - a -13 + b = 0

-a + b = 12 ---------(1)

put x = -3

(-3)³ -a(-3)² -13(-3) + b = 0

-27 -9a +39 + b = 0

-9a + b = -12 --------(2)

subtract equation (1) and (2)

-a + b + 9a - b = 12 + 12

8a = 24

a = 3

put a = 3 in equation (1)

b = 15

Answer 2

Sᴏʟᴜᴛɪᴏɴ :-

→ f(x) = x³ + ax² + bx - 45

→ Two Factors = (x - 1) & (x + 5) .

when (x - 1) is factor of f(x) :-

→ f(1) = x³ + ax² + bx - 45 = 0

→ (1)³ + a(1)² + b(1) - 45 = 0

→ 1 + a + b - 45 = 0

→ a + b - 44 = 0

→ a + b = 44 ---------- Equation (1)..

Similarly,

when (x + 5) is factor of f(x) :-

→ f(-5) = x³ + ax² + bx - 45 = 0

→ (-5)³ + a(-5)² + b(-5) - 45 = 0

→ (-125) + 25a - 5b - 45 = 0

→ 25a - 5b - 170 = 0

→ 5(5a - b) = 170

→ 5a - b = 34 ------------ Equation (2).

Adding Equation (1) & (2) now, we get,

→ (a + b) + (5a - b) = 44 + 34

→ a + 5a + b - b = 78

→ 6a = 78

→ a = 13 .

Putting value of a in Equation (1) now, we get,

→ b + 13 = 44

→ b = 44 - 13

→ b = 31 .

Hence, Value of a & b are 13 & 31 Respectively.

___________________

Now, Factorization :-

Given That, (x - 1) is a factor of given Polynomial .

So,

→ x³ + 13x² +31x - 45

→ x²(x - 1) + x² + 13x² + 31x - 45

→ x²(x - 1) + 14x² + 31x - 45

→ x²(x - 1) + 14x(x - 1) + 14x + 31x - 45

→ x²(x - 1 ) + 14x(x - 1)+45x - 45

→ x²(x - 1) + 14x(x - 1)+ 45(x - 1)

→ (x - 1)(x² + 14x + 45)

→ (x - 1)[ x²+ 5x + 9x + 45 ]

→ (x - 1)[ x(x+5) + 9(x+5) ]

→ (x - 1)(x+5)(x+9) (Ans.)