A polynomial f(x) with real coefficients and leading

Answered question

2022-04-01

A polynomial f(x) with real coefficients and leading coefficient 1 has the given zero and degree. Express f(x) as a product of linear and/or quadratic polynomials with real coefficients that are irreducible over ℝ.

7 + 8i;    degree 2

Answer & Explanation

user_27qwe

user_27qwe

Skilled2022-04-21Added 375 answers

The roots of the polynomial f(x) is

7+8i

Root is imaginary. The imaginary or complex Roots always exist as Complex Conjugates.

This implies, the second root is 7-8i and the two roots of f(x) are

7+8i, 7-8i

From Factor Theorem, if c is root of polynomial f(x) then (x-c) is a factor of the polynomial

Therefore, the polynomial f(x) can be written as

f(x)=(x-(7+8i)(x-(7-8i)

Simplify terms.

Simplify each term.

f(x)=(x-7-8i)(x-(7-8i))

Simplify each term.

Apply the distributive property.

f(x)=(x-7-8i)(x-17-(-8i))

Multiply -1 by 7.

f(x)=(x-7-8i)(x-7-(-8i))

Multiply -8 by -1.

f(x)=(x-7-8i)(x-7+8i)

Expand (x-7-8i)(x-7+8i) by multiplying each term in the first expression by each term in the second expression.

f(x)=xx+x-7+x(8i)-7x-7-7-7(8i)-8ix-8i-7-8i(8i)

Simplify terms.

Combine the opposite terms in xx+x-7+x(8i)-7x-7-7-7(8i)-8ix-8i-7-8i(8i).

f(x)=xx+x-7-7x-7-7-7(8i)-8i-7-8i(8i)

Simplify each term.

f(x)=x2-7x-7x+49-56i+56i+64

Simplify by adding terms.

f(x)=x214x+113

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