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A product of variables and numbers (like $3x$ or $5x^2$) is called a monomial. A sum of monomials (like $2x^2+4x+3$) is called a polynomial. An expression like $2x^2+4x+3$ is specifically called a quadratic polynomial because the highest power of $x$ is $x^2$. In the linear polynomials we studied earlier, the highest power of $x$ was $x^1$, i.e. just $x$. In this lesson, you will learn how to do arithmetic with quadratic polynomials.

Each row of the table below has a quadratic monomial or polynomial and a value of $x$. Compute the value of that expression at the given $x$.

$x$-value | expression | value of expression at $x$ |
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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. Your answer will be illustrated on the bottom grid.

If you have two quadratic polynomials (such as $2x^2+x+1$ and $x^2+2x+3$), you can add them by adding like terms (terms with the same power of $x$). So the sum of $2x^2+x+1$ and $x^2+2x+3$ is $3x^2+3x+4$, as shown below:

Add the two polynomials shown in each row of the table below. The sum is pictured on the grids to the left.

Similarly, you can subtract two quadratic polynomials by subtracting like terms.

Subtract the two quadratic polynomials shown in each row of the table below. The difference is pictured on the grids to the left.

As you can see, the sum or difference of two quadratic polynomials is another polynomial (either quadratic or linear).

To multiply a constant (such as 2) by a quadratic polynomial (such as $2x^2+x+5$), use the distributive law. For example:

$$ 2(2x^2+x+5)=2(2x^2)+2(x)+2(5)=4x^2+2x+10 $$as illustrated on the grids to the left.

Perform each multiplication problem in the table below, in the manner indicated in the first row of the table.

To multiply two linear polynomials together, you can also use the distributive law. For example, the top grid to the left shows $x(x+3)$ as the area of a rectangle with width $x$ and height $x+3$. The bottom grid shows how it can be expanded into $x(x)+x(3)=x^2+3x$.

Expand each of the products in the table below, as illustrated on the grid to the left.

In the previous question you used the distributive law once in each table row. If both of the polynomials you are multiplying have a constant term, you will need to use it more than once. For example, the top grid to the left is a picture of $(x+2)(x+3)$ as the area of a rectangle with width $x+2$ and height $x+3$. Using the distributive property, $(x+2)(x+3)$ becomes $x(x+3)+2(x+3)$, which is shown on the bottom grid.

Expand $x(x+3)$. |

Expand $2(x+3)$. |

Click to copy your two previous answers into the addition problem below.

Complete the expansion of $(x+2)(x+3)$ by doing this addition problem. |

Expand each of these products by using the distributive law, as in expandSlow.

You can also use this method to square linear polynomials. For example, $(x+2)^2=(x+2)(x+2)$, which can then be expanded as in expandFast:

$$(x+2)(x+2) = x(x+2)+2(x+2) = (x^2+2x) + (2x+4) = x^2 + 4x + 4$$Find the squares of the linear polynomials in the table below.