Practice Makes Perfect Algebra - Carolyn Wheater [15]
When monomials are combined by addition or subtraction, they form polynomials. A polynomial with two terms is a binomial, and one with three terms is a trinomial. For four or more terms, we use the general term polynomial.
Degree of a polynomial
The degree of a monomial containing one variable is the power to which the variable is raised. The degree of 3x5 is 5. The degree of x is 1, and the degree of any constant is 0. If a monomial contains more than one variable, for example, −6x2y3, its degree is the sum of the powers. The expression −6x2y3 is a fifth-degree monomial. The degree of a polynomial is the degree of its highest-power monomial. The degree of 6x4 − 3x2 + 12 is 4.
Standard form
A polynomial is in standard form when its terms are arranged in order from highest degree to lowest degree or lowest to highest degree. The polynomial 6x4 − 3x2 + 12 is in standard form, but −7t5 + 8t2 − 3t7 + 2t − 1 is not.
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EXERCISE 6.3
Put the polynomials in standard form and give the degree. If the expression is not a polynomial, explain why.
1. 5x + 3x2 −7 + 2x3
2. t7 −1 + 5t12 + 8t2 −9t
3. 5y6 −2y3 + 8 −12y11
5. 2x5 − 4x3 + 3x
6. 4 − 3z7 + 8z −4z2
7. 7 − 3w + w5 −9w3
8. b2 −3b −4 −b4
10. 6 − 7y3 + 8y2 − 4y
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Adding and subtracting polynomials
To add polynomials, follow the rules for combining like terms. If the terms match in both variable and power, add the coefficients. Keep the variable portion unchanged. Unlike terms cannot be combined. You will sometimes see polynomials enclosed by parentheses, as in(5x3 − 9x2 + 7x − 4) +(2x4 − 8x3 − 5x2 + 3). These parentheses are just to define the polynomials and can be dropped when adding. Rearrange the terms to bring like terms together and simplify.
(5x3 − 9x2 + 7x − 4) +(2x4 − 8x3 − 5x2 + 3)
= 5x3 − 9x2 + 7x − 4 + 2x4 − 8x3 − 5x2 + 3
= 2x4 + 5x3 − 8x3 − 9x2 − 5x2 + 7x − 4 + 3
= 2x4 − 3x3 − 14x2 + 7x−1
When you are subtracting polynomials, the parentheses have significance. To subtract polynomials, it is possible to subtract term by term. For example, the subtraction(4y3 + 3y2 + 7) −(−2y3 − 8y2 + y − 2) can be thought of as
[4y3 -(-2y3)] + [3y2 −(−8y2)] +(0y − y) + [7 −(−2)]
It is usually simpler, however, to treat subtraction as adding the opposite. To subtract(4y3 + 3y2 + 7) −(−2y3 − 8y2 + y − 2), think of it as(4y3 + 3y2 + 7) plus the opposite of(−2y3 − 8y2 + y − 2). The opposite of(−2y3 − 8y2 + y − 2) is(2y3 + 8y2 − y + 2). In the problem(4y3 + 3y2 + 7) −(−2y3 − 8y2 + y − 2), imagine that the subtraction sign is distributed to all the terms in the second set of parentheses; then drop the parentheses and add.
(4y3 + 3y2 + 7) −(−2y3 − 8y2 + y − 2)
=(4y3 + 3y2+7) +(2y3+ 8y2 − y + 2)
= 4y3 + 3y2 + 7 + 2y3+ 8y2 − y + 2
= 4y3 + 2y3 + 3y2 +8y2 − y + 7 + 2
= 6y3 + 1ly2 − y + 9
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EXERCISE 6.4
Add or subtract the polynomials as indicated and give your answers in simplest form.
1.(13w2 − 9w + 8) +(w2 − 9)
2.(a2 − 7a + 5) +(a2 + 2a − 9)
3.(−13x2 + 43x − 27) +(4x2 − 2x + 3)
4.(3y2 − 17y + 34) +(−7y2 + 14y − 2)
5.(1 − 2b + b2) −(−3 − b − 3b2)
6.(2b2 + 6b − 5) +(8 − 9b + 2b2)
7.(9x2 − 7x + 5) −(−2x2 + 6x + 3)
8.(2x2 − 3x) −(4x2 + 4x − 2)
9.(2x2 − x + 5) −(5x2 − 2x + 3)
10.(5x2 − 7x + 2) −(3x2 + 9x − 1)
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Multiplying polynomials
All polynomial multiplication is built on multiplying monomials, but we have different rules for polynomials of different sizes to make the work more efficient.
Multiplying monomials
To multiply two monomials, first multiply the coefficients. If the variables are the same, use rules for exponents to simplify. If the variables are different, just write them side by side. The product(−3x5y2)(2x2y3) =(−3 · 2)(x5 · x2)(y2 · y3) = −6x7 y5, but the product(5a2)(2b3) =(5 · 2)(a2)(b3) = 10a2b3.
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EXERCISE 6.5
Multiply