Question

Which polynomial function has a leading coefficient of 1, roots â€"2 and 7 with multiplicity 1, and root 5 with multiplicity 2? f(x) = 2(x 7)(x 5)(x â€" 2) f(x) = 2(x â€" 7)(x â€" 5)(x 2) f(x) = (x 7)(x 5)(x 5)(x â€" 2) f(x) = (x â€" 7)(x â€" 5)(x â€" 5)(x 2).

Answer 1

The correct option is d.

Which polynomial function has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2?

a) f(x) = 2(x + 7)(x + 5)(x – 2)

b) f(x) = 2(x – 7)(x – 5)(x + 2)

c) f(x) = (x + 7)(x + 5)(x + 5)(x – 2)

d) f(x) = (x – 7)(x – 5)(x – 5)(x + 2)

Leading coefficient: It is the coefficient of the term which is having the  highest degree in a given polynomial.

Now, to check the polynomial function that has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2;

a.)   f(x) = 2(x + 7)(x + 5)(x – 2),

Here the leading coefficient is 2.

Therefore, not the correct option.

b.)   f(x) = 2(x – 7)(x – 5)(x + 2),

Here the leading coefficient is 2.

Therefore, not the correct option.

c.)   f(x) = (x + 7)(x + 5)(x + 5)(x – 2),

Here the leading coefficient is 1.

Checking for roots for multiplicity,

$f(x)=(x + 7)(x + 5)(x + 5)(x-2)\\f(x)=(x + 7)(x + 5)^2+(x-2)$

x = -7, -5 and 2.

The roots are 2 and -7 with multiplicity 1, and root - 5 with multiplicity 2.

Therefore, not the correct option.

d.)   f(x) = (x – 7)(x – 5)(x – 5)(x + 2)

Here the leading coefficient is 1.

Checking for roots for multiplicity,

$f(x)=(x -7)(x-5)(x-5)(x + 2)\\f(x)=(x -7)(x-5)^2(x + 2)$

x = 7, 5 and -2.

The roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2.

Hence, the correct option is d.

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