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Chapter 0: Problem 11

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(17 x^{3}-5 x^{2}+4 x-3\right)-\left(5 x^{3}-9 x^{2}-8 x+11\right)$$

Short Answer

Expert verified
The resulting polynomial in standard form is \(12x^{3} + 4x^{2} + 12x - 14\) and its degree is \(3\).

Step by step solution

01

Identify Like Terms

In the given expression, identify the like terms in both the polynomials. Like terms are those terms which have the same variable raised to the same power. In the expression, \((17 x^{3}-5x^{2}+4x-3)\) and \((5 x^{3}-9 x^{2}-8 x+11)\), the like terms are \(17x^{3}\) and \(5x^{3}\), \(-5x^{2}\) and \(-9x^{2}\), \(4x\) and \(-8x\), and \(-3\) and \(11\).
02

Subtract Like Terms

Subtract the like terms from the polynomials. Hence, \(17x^{3} - 5x^{3} = 12x^{3}\), \(-5x^{2} - (-9x^{2}) = 4x^{2}\), \(4x - (-8x) = 12x\), and \(-3 - 11 = -14\). Hence, the subtraction of the given polynomials yields \(12x^{3} + 4x^{2} + 12x - 14\).
03

Write in Standard Form

The standard form for a polynomial arranges the terms by degree in descending order. However, the resulting polynomial \(12x^{3} + 4x^{2} + 12x - 14\) is already in standard form.
04

Determine Degree of Polynomial

The degree of the polynomial is determined by the largest exponent in the polynomial. In \((12x^{3} + 4x^{2} + 12x - 14)\), the largest exponent is \(3\). Therefore, the degree of the polynomial is \(3\).

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