Multiplying Polynomials

Let's find the required product using the Distributive Property. To start, we distribute the left-hand side binomial to each term of the right-hand side binomial. ( x-4)( x+2) ⇓ ( x-4) x+( x-4) 2 Next, we apply the Distributive Property once more to clear the parentheses and continue combining like terms to simplify the resulting expression.

To find the second product, we will use the FOIL Method. Remember, the word FOIL stands for First, Outer, Inner, and Last.

With this in mind, we can proceed with the required multiplication. Let's start by multiplying the first terms. ( y-3)( y+3) ⇓ y* y The empty box is there as a reminder that there are still missing terms. Next, we multiply the outer terms. ( y-3)(y+ 3) ⇓ y* y + y* 3 Now, it is the turn of the inner terms. (y- 3)( y+3) ⇓ y* y + y* 3 - 3* y Then, we multiply the last terms. (y- 3)(y+ 3) ⇓ y* y + y* 3 - 3* y - 3* 3 Finally, we perform each of the products and combine like terms to simplify the resulting polynomial.

For the third product, let's use the Box Method. First, we determine the dimensions of the table. To do this, let's count the number of terms in each polynomial. ccc Polynomial & & Number of Terms [0.1cm] (z+2) & & 2 [0.1cm] (1-z) & & 2 Since each polynomial has two terms, we will need a table with two rows and two columns. However, we will add an extra row and column to write the terms of each polynomial. Let's write the terms of the first polynomial in the first column and the terms of the second polynomial in the first row.

Next, we fill in the table's cells by multiplying the terms written on the corresponding borders of the table.

Finally, let's add all the expressions inside the table and combine like terms.

To start, let's write the formula for finding the area of a rectangle. A = l * w In our case, the rectangle has dimensions 3x+1 and x-4. Let's substitute them into the formula. A(x) = (3x+1)(x-4) Finally, we write the previous polynomial in standard form by multiplying the two binomials.

The last polynomial represents the area of the given rectangle.

We are interested in finding the area of the given triangle. Therefore, let's begin by writing the formula that we will use. A = 1/2bh Since the given triangle is a right triangle, we can use the legs to find the area — that is, b=4g+1 and h=9g-2. Let's substitute these expressions into the formula. A(g) = 1/2(4g+1)(9g-2) To write the polynomial A(g) in standard form, we can multiply the two binomials using the FOIL Method.

Note that the area of the shaded region is the area of the rectangle minus the area of the square. A_(Shaded) = A_R - A_S Let's start by finding the area of the rectangle. To do this, we multiply its dimensions.

Next, let's find the area of the square.

Finally, let's subtract the square's area from the rectangle's area.

The obtained polynomial represents the area of the given shaded region.

Let's recall the formula for the surface area of a cube. SA =6l ^2 Here l is the side length of the cube. In our case l= 2k+1. Let's substitute this expression into the formula. A(k)=6( 2k+1)^2 Now, let's simplify the obtained expression and write it in standard form.

We can multiply the binomials using the FOIL Method. Recall that the word FOIL stands for First, Outer, Inner, and Last.

Finally, we combine like terms to simplify the polynomial.

The surface area of the given cube is represented by A(k)=24k^2 + 24k + 6.

Let's disassemble the given rectangular prism to see the faces individually.

With this in mind, we can write an expression representing the surface area of the prism. SA = 2A_1 + 2A_2 + 2A_3 ⇓ SA = 2(A_1 + A_2 + A_3) Let's start by finding the area of the bases, whose dimensions are 2p and 3p+1.

In a similar way, let's find the area of the lateral face with dimensions 3p+1 and 9p-5.

Next, let's find the area of the lateral face with dimensions 2p and 9p-5.

Now we are ready to find the polynomial representing the surface area of the rectangular prism. To do this, we substitute the expressions we found for A_1, A_2, and A_3.

According to the box method, the top row labels are the terms of one of the polynomials being multiplied. Similarly, the elements at the left of the table form the other polynomial. Let's start by writing the polynomials being multiplied. P(x) &= x^2-2x+6 Q(x) &= 2x^2+3 Now, the cells of the table are filled by multiplying the terms written on the corresponding borders of the table. For example, the top-left cell corresponds to the product of 2x^2 and x^2.

In a similar fashion, the second cell of the second row corresponds to the product of 3 and -2x.

Knowing this, we can find A and B. A &= 2x^2* x^2 = 2x^4 B &= 3(-2x) = -6x Finally, we add the two previous expressions. A+B = 2x^4 - 6x

Now that we know the expressions for A and B, let's begin by writing the in the table.

The result of the polynomial multiplication is the sum of all the expressions inside the table. Therefore, let's add them and combine like terms, if any.

The degree of a polynomial is the highest exponent in the polynomial. Therefore, let's start by finding R(x).

From the resulting polynomial, we can see that the highest exponent is 9. Therefore, the degree of R(x) is 9.

The leading coefficient of a polynomial is the coefficient in front of the term with the highest exponent. Knowing this, let's highlight the leading coefficients of P(x), Q(x), and R(x). P(x) &= 2x^3-5 Q(x) &= 4x^6+2 R(x) &= 8x^9 - 20x^6 + 4x^3 - 10 The leading coefficients of P(x), Q(x), and R(x) are 2, 4, and 8, respectively. Let's find the required sum using this information! 2 + 4 + 8 = 14

The constant term of a polynomial is the term with no variables including its sign. R(x) = 8x^9 - 20x^6 + 4x^3 - 10 Therefore, the constant term of R(x) is -10.