More Quadratic Polynomials

Typing entire polynomials

For each row of the table below, a picture illustrating a quadratic polynomial will appear on the top grid to the left. Slide the $x$ slider to determine what polynomial it is, and enter that polynomial into the table. Use the ‘^’ key to type in powers. For instance, type in $x^2$ as “x^2”. Write higher powers of $x$ before lower ones.

Multiplying linear polynomials

In the previous lesson, we multiplied together linear polynomials like $x+2$ and $x-3$, whose coefficients of $x$ were 1. We will now look at multiplying together linear polynomials with other coefficients of $x$.

Find the indicated products, as suggested by the grid to the left.

You can multiply together more complicated linear expressions by using the distributive law, in much the same way as when the coefficient of $x$ is 1. For example, $2x(3x+1)$ can be expanded as follows:

$$ 2x(3x+1) = 2x(3x)+2x(1) = 6x^2 + 2x $$

Expand each of the expressions below. (Remember to type just $x$ instead of $1x$.)

Expand each of the expressions below, as shown in the example row.

Multiple simplifications

If you want to expand an expression like $2(x+4)(3x+2)$, there are two different ways you can do it.

Either multiply the constant factor by the first linear factor: $$ \cl"tight"{\table , 2(x+4)(3x+2); =, (2(x)+2(4))(3x+2); =, (2x+8)(3x+2)} $$ and then expand the result as you did in nonHomogQuestion: $$ \cl"tight"{\table , \colspan 7 (2x+8)(3x+2); =, \colspan 3 2x(3x+2), +, \colspan 3 8(3x+2); =, 2x(3x), +, 2x(2), +, 8(3x), +, 8(2); =, 6x^2, +, 4x, +, 24x, +, 16; =, \colspan 7 \cl"highl"{6x^2 + 28x + 16}} $$

Or, multiply the two linear factors first: $$ \cl"tight"{\table , 2(x+4)(3x+2); =, 2(x(3x+2) + 4(3x+2)); =, 2(x(3x) + x(2) + 4(3x) + 4(2)); =, 2(3x^2 + 2x + 12x + 8); =, 2(3x^2 + 14x + 8)} $$ and then expand this as you did in the previous lesson: $$ \cl"tight"{\table , 2(3x^2 + 14x + 8); =, 2(3x^2) + 2(14x) + 2(8); =, \cl"highl"{6x^2 + 28x + 16}} $$

In either case, the final answer will be the same.

Expand each of the expressions below, using either of the methods above. Use scratch paper.

By combining the techniques you have learned in this lesson and the previous one, expand and simplify each of the expressions below. Your final answer should be of the form $ax^2+bx+c$.