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# Polynomials

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### Polynomials

1. 1. Polynomial s What is POLYNOMIAL? An expression which is the sum of terms of the form a x k where k is a nonnegative integer is a polynomial. Polynomials are usually written in standard form.
2. 2. Polynomial s Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. Polynomial in standard form: Degree 2 x 3 + 5x 2 – 4 x + 7 Leading coefficient Constant term
3. 3. Polynomial s The degree of each term of a polynomial is the exponent of the variable. The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient.
4. 4. Polynomial s Kinds of Polynomials (according to number of terms) A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial.
5. 5. Adding Polynomials Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3) SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x 3 + 2 x 2 – x + 7 3x 2 – 4 x + 7 + – x + 4x –8 3 2 4x 3 + 9x 2 – 5x + 6
6. 6. Adding Polynomials Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) SOLUTION Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) = (2 x 2 + x 2) + (x + x) + (–5 + 6) = 3x 2 + 2 x + 1
7. 7. Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 No change –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 Add the opposite + 2 x3 – 3x + 4
8. 8. Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 + 2 x3 – 3x + 4 5x 2 – 4x + 12
9. 9. Subtracting Polynomials Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) SOLUTION Use a horizontal format. (3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2) + (– 5x + x) + (3 + 4) = x 2 – 4x + 7
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