Once you have terms, it’s natural to want to start doing things with them. You can add or subtract terms if they’re like terms, and you can multiply terms by other terms, like or not. You can even do both of these things, adding and then multiplying. When you start doing so many things with terms, you soon need to stop and think about how you simplify the problems you’ve created.
Do you need a whole new set of rules? The good news is no. Your old friend PEMDAS
(Parentheses, Exponents, Multiply, Divide, Add, Subtract) will cover most situations. Let’s look at a problem that’s not too complicated. Suppose you wanted to do 5x(3x + 7x). PEMDAS says do what’s in the parentheses first, so look at the 3x + 7x part first. Those are like terms, so you can add them and get 10x. Your expression 5x(3x + 7x) becomes 5x(10x). Exponents are next, but there are none, so move on to multiplying and dividing. 5x(10x) = 5 × 10 × x × x = 50x2.
How about this one? Simplify -4xy(6x + 7y) using the same rules. Parentheses first. But this time, you can’t add what’s in the parentheses because they’re unlike terms. Don’t panic. You can only do what you can do, so just move on. There are no exponents, so it’s time to multiply, and for that, you’ll need the distributive property.
-4xy(6x + 7y) = -4xy(6x) + (-4xy)(7y)
Now, as you look at each multiplication, multiply coefficients and combine what you can. Don’t be afraid to rearrange. Remember multiplication is commutative and associative.
-4xy(6x) = -4 × 6 × x × x × y = -24x2y (-4xy)(7y) = -4 × 7 × x × y × y = -28xy2 Put the pieces back together to get
-4xy(6x + 7y) = -4xy(6x) + (-4xy)(7y) = -24x2y + -28xy2
Because you have the plus from the addition problem followed immediately by the minus sign from the -28, you can write your answer as -24x2y – 28xy2.
Your basic rules are:
• Combine what’s in the parentheses if you can.
• Distribute multiplication over the addition or subtraction if you can’t combine the unlike terms.
• Simplify each multiplication.
• Check the signs.
Whenever you have negatives in problems like these, it’s important to be careful about the rules for signs. If the multiplier that you’re distributing is positive, the signs aren’t usually a problem.
Terms in the parentheses that were positive will produce positive terms in the answer, and terms that were negative in the parentheses will produce negative terms in the answer. Here are some examples.
8x(-3a + 2b) = 8 × (-3) × x × a + 8 × 2 × x × b = -24ax + 16bx 5t(7t2 – 3t) = 5 × 7 × t × t2 – 5 × 3 × t × t = 35t3 – 15t2
When the multiplier is negative, however, you need to work carefully and remember that if you distribute a negative term, the signs of each term in the parentheses will change. Let’s take those two examples and make the multipliers negative to see what happens.
Take the first example and change 8x to -8x. Distribute the -8x.
-8x(-3a + 2b) = -8 × (-3) × x × a + (-8) × 2 × x × b
Now that first term has a negative multiplied by a negative, so the result is going to be positive.
The second term has one negative, so that result will be negative.
-8x(-3a + 2b) = -8 × (-3) × x × a + (-8) × 2 × x × b = +24ax + -16bx You can eliminate the plus signs, condensing to 24ax – 16x.
If you change 5t to -5t in the second example and distribute, you get -5t(7t2 – 3t) = -5 × 7 × t × t2 – (-5) × 3 × t × t.
The first term will come out negative, but the second term is where you have to be careful. The product (-5) × 3 × t × t will give you a negative result, -15t2, but you also have a minus in front of that, the one that was connecting the original terms. You get a product of -35t3 – -15t2, but the double minus becomes a plus so you end up with -35t3 + 15t2. The minus on the multiplier switches the signs.
Sometimes you’ll find an expression with more than one set of parentheses and lots of terms, some like and some unlike, and some that start out unlike and then turn into like terms. You just need to take things step by step and pay attention to what is happening. Here’s an example.
-3x(6x2 – 5) + 8x2(4x – 9)
There are no like terms in the first parentheses, so you can’t do anything there, and no like terms in the second set of parentheses, either. Move on to multiplying. Use the distributive property.
-3x(6x2 – 5) + 8x2(4x – 9)
= -3 × 6 × x × x2 – (-3) × 5 × x + 8x2(4x – 9)
= -3 × 6 × x × x2 – (-3) × 5 × x + 8 × 4 × x2 × x – 8 × 9 × x2
= -18x3 + 15x + 32x3 – 72x2
Now notice you have like terms: -18x3 and 32x3. You can combine those to get a final answer of 14x3 + 15x -72x2. It’s traditional to put your terms in order from highest exponent to lowest, so rewrite it as 14x3 – 72x2 + 15x.
CHECK POINT
Simplify each expression.
16. 6x(2x + 9) 17. 12 + 5(x + 1) 18. 6t2(t – 3) – 2t2
19. 5y(6y + 2) + 7y2(4 – 12y) 20. 8a(2b – 5) – 2b(a – 2)
Polynomials
There’s a particular group of expressions, which you’ll hear about in algebra, that are called polynomials. The prefix poly means “many,” so it would seem that they would be expressions with many terms, and some of them are. But the name polynomial is applied to any expression that fits a particular pattern, even those with only one term.
The pattern is easier to show than to describe, but let’s try. Polynomials are expressions that are made by adding terms that are the product of a numerical coefficient and a power of a certain variable. For example, 8x5, -6x3, x2, and 2x are all terms that are the product of a numerical coefficient and a power of the variable x. You don’t see the numerical coefficient in x2 because it’s 1, and we don’t usually show that. Constants can also be part of a polynomial because we can say a constant, like 3, is 3x0.
Each of these terms could be called a polynomial all by itself. It would be a one-term polynomial, also called a monomial. Mono is the prefix that means “one.” You can add monomials with the same variable to make a more complex polynomial. For example, 8x5 + -6x3 + x2 + 2x + 3 is a polynomial. If you add two monomials, like 5y3 + 2y, you make a binomial. If you add three monomials, like t2 + 3t + 1, that’s a trinomial. Monomials, binomials, and trinomials are all types of polynomials.
The degree of a monomial is the exponent on the variable. 8x5 is fifth degree, x2 is second degree, and 2x is first degree. You don’t see an exponent of 1, but that’s what 2x really means: 2x1. Constants are degree zero, because we’re thinking of them as a constant times x0.
MATH TRAP
Be careful to look at all the terms of a polynomial before you decide on its degree.
Don’t just jump at the term that happens to be written first.
The degree of a polynomial is the highest degree of all its monomials. The polynomial 8x5 + -6x3 + x2 + 2x + 3 is a fifth degree polynomial, and t2 – 7t + 4 is a second degree poly-nomial. For the polynomial 5t3 -7t + 8t4 -2t2 + 5, you need to be careful to look at the whole polynomial. The degree of the polynomial is 4, not 3. The highest degree term is in the middle, not the beginning, of the polynomial.
If there is more than one variable in a term, the degree of the term is the sum of the degrees of each variable. The term xy is degree 2, one for x and one for y. The term 3x4y2 is degree 6.
CHECK POINT
Give the degree of each polynomial.
21. -3a3 + 5a2 – 3a + 12 22. 6 + 3b -4b2
23. 2t – 9 + 7t2 +4t3
24. 11y – 7y4 + 5y2 – 3 25. 6 – 4x2 + 3x
When you write a polynomial, no matter how many terms it has, it’s traditional to write it in standard form. Standard form means that you write the terms of the polynomial in order from the highest degree monomial to the lowest. The polynomial 8x5 + -6x3 + x2 + 2x + 3 is in standard form because the monomials start with the fifth degree, then the third degree, then the second, first, and finally the zero degree term.
WORLDLY WISDOM
The standard form of a polynomial writes the terms in order from highest degree to the lowest.
The polynomial 4t3 – 6t9 + 11t – 5t2 + 8t7 is not in standard form. To put it in standard form, you need to first locate the highest degree term. In this case, that’s -6t9. That has to be first, so move it around to the front. The 4t3 that was in front is positive, so you can put a plus between the ninth degree term and the 4t3. -6t9 + 4t3 + 11t – 5t2 + 8t7.
There’s no eighth degree term, but there’s a seventh degree, so that should be next.
-6t9 + 8t7+ 4t3 + 11t – 5t2.
There are no sixth, fifth, or fourth degree terms, and the third degree term is already next in line. You just need to swap the last two terms. -6t9 + 8t7 + 4t3 – 5t2 + 11t is in standard form.
CHECK POINT
Put each polynomial in standard form.
26. a3 + 10a4 – 11a + 9 27. 2b3 – 9b + 12b2 – 5 28. 3k4 + 8k5 – 13k – 7
29. 4 – 7m2 + 14m4 + 2m3 30. 5p – 3 + 6p2 – 15p3
The Least You Need to Know
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A term is a constant, a variable, or the product of a constant and a variable.•
Terms are “like” if they contain exactly the same variables, raised to the same power, such as 2y and 10y.•
Only like terms can be combined.•
To add or subtract like terms, add or subtract the coefficients and keep the vari-able part the same.•
A polynomial is a sum of terms using different powers of the same variable.A monomial has one term, a binomial has two terms, and a trinomial has three terms.