Polynomials and Exponents
Mathematics, Grade 8
Polynomials and Exponents
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Study Guide Polynomials and Exponents Mathematics, Grade 8
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POLYNOMIALS AND EXPONENTS A polynomial is an expression which is in the form of axn, where a is any real number and n is a whole number. • If a polynomial has only one term, it is called a monomial. If it has two terms, it is a binomial and if it has three terms, it is a trinomial. • The standard form of a polynomial is when the powers of the variables are decreasing from left to right. Mathematical operations can be performed on polynomials. • To add or subtract polynomials, like terms are combined. • To multiply polynomials, one polynomial is distributed to the other polynomial. Since polynomials contain variables raised to a power, the rules for exponents must be followed. • When adding or subtracting, only like terms are combined, so the exponents do not change. • When multiplying variables with exponents, the exponents are added together. Scientific notation uses the rules for multiplication of exponents because scientific notation is used to write very large or very small numbers by using the power of ten. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
How to use polynomials and exponents Polynomials can be added or subtracted by combining like terms. Ex. (6x² + 2x) + (8x² - x) = 14x² - x • When adding or subtracting, the variables stay the same and the coefficients are added or subtracted. When subtracting polynomials, it is important to distribute the negative sign to the second polynomial. Ex. (4x² + 5x - 3) - (3x² - 2x) = 4x² + 5x -3 - 3x² + 2x Notice how in the second term the 3x² and 2x changed signs because of distributing the negative. The polynomial can be added by combining like terms and the result is x² + 7x - 3. • Polynomials can also be added and subtracted vertically, but again the negative sign must be distributed with subtraction. • To multiply polynomials, one polynomial is distributed to another polynomial. For example, if the binomial, 2x², is multiplied by the trinomial, 2x² + 3x + 4 the result would be as follows: (2x²)(2x² + 3x + 4) = 4x4 + 6x³ + 8x² Notice the coefficients are multiplied and the variables are multiplied. The exponents are added together when multiplication is preformed on a variable with the same base. • When a variable with the same base is divided, the exponents are subtracted. Any number or variable to the zero power is 1. • Polynomials can also be multiplied by a method called FOIL. This is used when multiplying two binomials. Ex. (3x² + 5)(x² - 6) = (3x²)(x²) + (3x²)(-6) + (5)(x²) + (5)(-6) = 3x4 - 18 x² + 5x² - 30 F O I L The term FOIL refers to the order in which the terms are multiplied: first, inside, outside, last. When the middle terms are combined, the final result is 3x4 - 13x² - 30. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
• When using scientific notation, the numbers are multiplied together separately from the powers. Ex. (1.2 x 10²) x (2.4 x 105) = 2.88 x 107 Try This! 1. Add the polynomials: (5x² + 2x + 6) + (3x² + 4x + 7) 2. Subtract the polynomials: (-8x³ + 3x² + 5) - (4x³ + 2x² - 8) 3. Multiply the polynomials: (6x)(x³ - 4x² - 12) 4. Multiply using FOIL: (2x - 6)(3x + 7) 5. Solve: 60 = ___? x³ · x7 = ___? 97/93 = ___? 3-2 = ___? 6. Multiply: (3.4 x 105)(2.3 x 10²) © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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