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Video Transcript (0:06) Before we learn how to multiply polynomials together we're going to solve a simpler problem. Here are four rectangles, one of them is 10 by 10, one of them is 7 by 10, one of them is 2 by 10 and one of them is 7 by 2. (0:20) Can we find the area of these rectangles? Of course we can 10 * 10 is 100, 7 * 10 is 70, 2 * 10 is 20, and 7 * 2 is 14. (0:30) You may have already noticed but these four smaller rectangles are combined into one big rectangle which is 17 by 12 and one way to calculate the area of the big rectangle would be to multiply 17 * 12 but another way would be to add up all of the smaller rectangles that we have. 100 + 70 + 20 + 14 makes 204 and 17 * 12 equals 204. (1:01) So what's the point? Why are we talking about this in a video on multiplying polynomials? well the way we just solve this problem is the same way we can multiply polynomials. what if we replaced those tens with “x”s. (1:18) Now the sides of our rectangle are x + 7 and x + 2 So the area would be (x + 7) * (x + 2). and we can work that the way that we just did - by looking at the parts. x times x is x squared, 7 * x is 7x, 2 * x is 2x and 7 * 2 is 14. so if we put those together we can figure out that (x + 7) * (x + 2) is x^2 + 7x + 2x + 14 which simplifies to x^2 + 9x + 14. (1:53) Now there's a way to remember this process which is called FOIL. FOIL is an acronym which stands for first, outside, inside, last. (2:03) The way that it works is it helps you remember what to multiply first to do the first terms in each parentheses which would be x * x ,then the two outside terms which would be 2 * x then the two inside terms which would be 7 * x and then the last two terms in each parentheses which would be 7 * 2. And if we put those together we get the same answer. (2:25) Now this is a really helpful way to remember everything, but we shouldn't lose sight of what's really going on. the fundamental idea is we multiply each of the terms in the first (by each of the terms in the second) and then we combine like terms. and it's important to understand this process for solving difficult problems. Let's look at one. (2:49) Let’s multiply (x^2 - 2x + 3)(x^2 + 5x - 2) (2:56) FOIL doesn't help us here - it only works for two terms times two more terms. Let's look at a picture for this one more time. We’ll make one big rectangle and break it up into all of the parts. (3:10) So we're going to have 9 terms to deal with. For the top row we get x^2 * x^2 = x^4, x^2 * -2x = -2x^3 and x^2 * 3 = 3x^2. (3:24) On the next row we get 5x * x^2 = 5x^3, 5x * -2x = -10x^2, 5x * 3= 15x. (3:36) On the last row we get -2 * x^2 = -2x^2, -2 * -2x = 4x and -2 * 3 = 6. (3:50) Now we just need to collect like terms and they're actually already organized for us to look at the diagonals. (4:00) So we get x^4 + 3x^3 -9x^2 +19x - 6, (4:30) And we’re done. And that's all there is to it when multiplying polynomials we just multiply all of the terms of 1 polynomial by all of the terms in the other and simplify.
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Video Transcript (0:07) What are polynomial factors? (0:10) The factors of any number are simply the numbers that multiply together to equal that number. For instance, 4 is a factor of 24, since 4 × 6 = 24. (0:22) For polynomials, factors are typically other polynomials! A polynomial could look like x2 - x - 6, which has two factors. (0:32) The two factors are (x - 3) and (x + 2). We can verify this by multiplying these two factors together, using the FOIL method. (0:43) The FOIL method is a mnemonic device to remember how to multiply binomials together. It stands for First Outer Inner Last (0:52) To multiply (x - 3)(x + 2), multiply the first of each binomial to find x × x = x2 . Multiply the outer terms to find that x × 2 = 2x . Multiply the inner terms to find that -3 × x = -3x. Finally, multiply the last terms to find that -3 × 2 = -6. (1:18) Combine the middle two terms and add all of this together to find the polynomial x2 - x - 6. (1:26) Factoring by itself is a very useful skill, but this lesson will focus more on what we can use the polynomial factors to do. (1:35) Most importantly, polynomial factors tell us the x-intercepts, zeros, or roots of our polynomial. All three of these names mean the same thing: where our polynomial crosses the x-axis. (1:49) In the previous example we found that x2 - x - 6 = (x - 3)(x + 2). Since we are interested in the graph of this polynomial, we can set it equal to and see the following graph. (2:04) Notice that this quadratic graph crosses the x-axis at (-2, 0) and (3, 0). (2:11) This seems a bit odd as the polynomial factors (x - 3) and (x + 2). It feels a bit like these x-intercepts should be -3 and positive 2. (2:23) What’s happening here is that since we are considering the graph y = (x-3)(x+3), we are trying to determine where y = 0. Plugging this into the equation, we see that 0 = (x-3)(x+3). (2:39) This means that one of two things is true. Either x - 3 = 0 or x + 2 = 0. Take a moment and consider why that is. (2:52) Notice that since the left hand side of the equation is 0, we know one of the two factors must be 0 on the right hand side. (3:00) Solve both of these equations for x to find that x = 3 or x = -2, which is exactly what we discovered from the graph alone! (3:09) This is why polynomial factors are so powerful. They immediately show us where the zeroes of our polynomial are. (3:17) If I told you that some polynomial p(x) had a factor of 4x + 3, you wouldn’t be able to tell me much about the polynomial, but we would know that when 4x + 3 = 0 we could find that x = -3/4 and know that this is one of the -intercepts of our graph! (3:38) The other major use of polynomial factors is determining the behavior of our graph near the x-axis. (3:45) For example, the graph of y = (x - 2)2(x - 2)3(x + 1) looks something like this: (3:55) Notice that all of the x-intercepts are -1, 1 and 2, just like we would predict. (4:02) However, we can be more specific with how the graph is behaving at each of these points. (4:08) The x-intercept at -1 is the most straightforward. Since this corresponds to the polynomial factor (x + 1) , which is not raised to any power, the graph simply crosses the x-axis at x = -1. (4:22) The x-intercept at 2 is the second most straightforward. Notice that this corresponds to the polynomial factor (x - 2)2. Since this polynomial factor is squared, the graph acts like a parabola and only touches our x-axis, not crossing it! (4:39) This is what will happen for every polynomial factor that is squared. (4:45) Finally, we can consider the x-intercept at 1. This corresponds to the polynomial factor (x - 1)3 and it crosses the x-axis, but it behaves a bit strangely. Notice that the graph initially looks like it will not cross, but then it does cross the x-axis. (5:02) This is because this is how the graph of y = x3 behaves as well! When you see this behavior, it is due to a cubed polynomial factor. (5:13) Ultimately, polynomial factors are a powerful and versatile tool that allow you to understand the zeros of polynomials and their graphs in much greater detail.
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Algebraic expressions can be made up of one or more terms, which are either added or subtracted from each other. These are known as polynomials and can include one or more variables, their exponents, and known numbers called constants. Definition of a Polynomial and Classifications Contains 1 or more terms in the expression d is the coefficient for the variable 𝑥3 a is the coefficient for the variable 𝑥2 b is the coefficient for the variable 𝑥 c is the constant (number: 4, 10, -5, etc.) Keep In Mind Coefficients and the constant, c, can be ANY real number. When listing the variables, ANY variables with their own distinct exponents can be used in place of 𝑥3, 𝑥2, 𝑥. (i.e. xy2, f2, g3, h7, z, etc. A lone constant (i.e. 2, -15, 1.4) CAN be classified as both a polynomial AND a monomial. Polynomials with more than 4 terms exist but are unlikely to be tested on the ISEE. While true polynomials always contain positive, whole number exponents, the ISEE can include rational (fractional) exponents. Depending on their number of terms, a polynomial can be further classified as a monomial, binomial, or trinomial. Multiplying Polynomials Each of the following methods is a variation of the distributive property. Choose the method that is most intuitive for you; the following examples will focus on multiplying binomials. Multiplying Polynomials on the ISEE Quadratics are fairly common on the ISEE so expect to see multiplication of binomials. When you encounter polynomial expressions that have been raised to an exponent, write out the terms and avoid this common mistake: Correct Incorrect (a + b)2 = (a + b)(a + b) (a + b)2 ≠ a2 + b2 (a + b)3 = (a + b)(a + b)(a + b) (a + b)3 ≠ a3+b3 Problems that involve multiplying polynomials with two or more terms OR multiplying more than two polynomials can also be tested. Continuing the example from above: (a + b)3 = (a + b)(a + b)(a + b) Some questions may ask for the coefficients of one or more variables several different ways. The ISEE often integrates multiple topics together on test questions. Some problems may ask you to incorporate polynomial multiplication when calculating the area or volume of geometric figures. Practice Problems Answers to Practice Questions D A C B A
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