(a) (2 x+3)(5 x-4) (b) 5 x(2 x+1)+2(x-6) (c) 2(x-2)(5 x+3) (d) 7 x(x+1)+3(x-2)(x+2) (e) 10(x^2-1

(a) (2 x+3)(5 x-4) (b) 5 x(2 x+1)+2(x-6) (c) 2(x-2)(5 x+3)...

Key Concepts

Difference of Squares

The difference of squares is a specific factoring pattern where an expression takes the form a² ? b², which can be rewritten as (a ? b)(a + b). Recognizing this pattern allows for quicker simplification and factoring of polynomials, especially when dealing with expressions that subtract one square from another.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by summing coefficients of terms that have the same variable raised to the same power. This is essential after performing operations like distribution or binomial multiplication, as it reduces the expression to its simplest form.

Multiplication of Binomials (FOIL Method)

Multiplying binomials involves using the FOIL method—First, Outer, Inner, Last—to ensure that each term in the first binomial is multiplied by each term in the second binomial. This technique is key for expanding products of binomials, leading to quadratic expressions that can then be simplified by combining like terms.

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by each term within a parenthesis. This property is used extensively in expanding expressions and simplifying algebraic equations by ensuring that every term in one expression multiplies every term in the other when necessary.

Transcript

00:01 Here in this video, we have a series of problems that requires us to either use polynomial multiplication or polynomial division, which is also known as long division.

00:12 We're going to start from a going all the way down to f, where the first problem requires us to use polynomial multiplication.

00:21 Now, when you're doing polynomial multiplication, there are three steps that you need to follow.

00:26 The first one is you need to multiply each term in one polynomial.

00:30 By each term in the other polynomial using the distributive law.

00:36 The second step is you need to add the powers of the same variables using the exponent rule, which you'll see on the right hand side where there's a little diagram.

00:46 That is the product rule within the laws of the exponents.

00:50 And the final rule is you need to simplify the resulting polynomial by adding or subtracting the like terms.

00:58 So i'm going to start by looking at the right term.

01:01 First with 4x, and i'm going to distribute that to each term on the left -hand side.

01:08 So 4x times 2x squared is 8x to the third, 4x times 5x is 20x squared, 4x times negative 3 is negative 12x.

01:25 Now i'm going to look at the second term, which is negative 7, and i'm going to multiply that by each term in the left -hand side.

01:32 So negative 7 times 2x squared is negative 14 x squared, negative 7 times 5x is negative 35x, and negative 7 times negative 3 is 21.

01:53 Now we're going to simplify this resulting polynomial by looking at the highest power to the lowest power.

02:00 So if we do that, we get 8x to the third because that's the only term that has a power to the third.

02:09 Plus 6x squared because we're adding the two terms that have a second power, minus 47x, adding the two terms that have a first power, and finally, plus 21.

02:32 And that is the final answer to this problem.

02:35 Now we're looking at problem b, which is again polynomial multiplication, and we're going to apply the same steps.

02:41 So first looking at the right -hand side with 5x, we're going to distribute that to each term on the left -hand side.

02:50 So 5x times 4x squared is 20x to the third, 5x times negative 7x is negative 35x squared, 5x times 3 is 15x.

03:07 Looking at the next term, 6, we're going to distribute that to each term on the left -hand side as well.

03:13 Where 6 times 4x squared is 24x squared, 6 times negative 7x is negative 42x, and 6 times 3 is 18.

03:27 And by simplifying this resulting polynomial, starting from the highest power to the lowest, our answer is 20x to the 3rd minus 11x squared minus 27x.

03:47 Plus 18.

03:49 And that's the final answer for this problem.

03:53 Let's now take a look at problem c, starting with a term on the right hand side, with 3x, and distributing that to each term on the left hand side.

04:03 By doing that, our first half of the equation is 15x to the third minus 9x squared minus 12x.

04:18 Going to the next term, negative 5, and distributing that to each term on the left hand side as well.

04:24 Our second, half of the equation is negative 25x squared plus 15x plus 20.

04:37 And by simplifying this polynomial, our final answer is 15x to the third minus 34x squared plus 3x plus 20.

04:57 That's our final answer.

05:00 Now we're looking at a problem that requires us to use polynomial division, but in this case, we're going to solve it using long division.

05:10 When you solve these kinds of problems, there are three steps that you need to follow.

05:14 The first one is you need to set up this problem in long division form, which i'll show you in just a second.

05:19 If there are terms that are missing, you need to replace those terms with zeros.

05:24 Secondly, you then need to ask yourself what you need to multiply the divisor by to get the first term in the dividend, and finally, you multiply that term or result by the divisor and subtract from the dividend.

05:38 If we set this up in long division form, our problem will then look like this.

05:57 To solve this, you need to ask yourself, what do i need to multiply this divisor by in order to get this first term in the dividend, which is 6x to the 3rd? the answer to that is 3x squared...

Please give Ace some feedback