Lesson 1.4: Polynomials

New Concept

Polynomials are expressions built from variables raised to non-negative integer powers. This lesson covers identifying their key features (degree, coefficients) and performing fundamental operations: addition, subtraction, and multiplication.

What’s next

You're now ready for some examples. Next, you'll work through interactive cards on adding, subtracting, and multiplying polynomials to master these essential skills.

Property

A polynomial is an expression that can be written in the form

Examples

• For $5x^4 - 3x^2 + 7$, the degree is $4$, the leading term is $5x^4$, and the leading coefficient is $5$.
• For $8 + y - 6y^3$, the degree is $3$, the leading term is $-6y^3$, and the leading coefficient is $-6$.
• For $z^7 - 9z^5 + z$, the degree is $7$, the leading term is $z^7$, and the leading coefficient is $1$.

Explanation

A polynomial is a sum of terms where variables have non-negative integer exponents. The degree is the highest exponent, which indicates the polynomial's complexity. The leading term and its coefficient are important for understanding the polynomial's behavior.

Property

To add and subtract polynomials, combine like terms, which are terms that contain the same variables raised to the same exponents.
Then, simplify and write the resulting polynomial in standard form. For subtraction, this is equivalent to adding the opposite of the second polynomial to the first.

Examples

• To find the sum of $(5x^2 + 2x - 10) + (2x^2 - 7x + 3)$, combine like terms: $(5x^2+2x^2) + (2x-7x) + (-10+3) = 7x^2 - 5x - 7$.
• To find the difference $(8y^3 - 3y + 4) - (2y^3 + y^2 - y)$, distribute the negative: $8y^3 - 3y + 4 - 2y^3 - y^2 + y$. Then combine like terms to get $6y^3 - y^2 - 2y + 4$.
• To find the difference $(-4a^2 + 5) - (a^3 - 4a^2 - 3)$, distribute the negative to get $-4a^2 + 5 - a^3 + 4a^2 + 3$. Combine like terms to simplify to $-a^3 + 8$.

Explanation

Think of this like sorting fruit. You can only combine apples with apples ($x^2$ with $x^2$), not with oranges ($x$). When subtracting, remember to distribute the negative sign to every term in the second polynomial before combining.

Property

To multiply polynomials, use the distributive property to multiply each term of the first polynomial by each term of the second.
After multiplying, combine all the resulting like terms and simplify to get the final expression.

Examples

• To find the product of $(3x-2)(4x^2+x-5)$, distribute $3x$ and $-2$: $3x(4x^2+x-5) - 2(4x^2+x-5) = 12x^3+3x^2-15x - 8x^2-2x+10$. Combining like terms gives $12x^3 - 5x^2 - 17x + 10$.
• To find the product of $5y(2y^2 - 3y + 1)$, distribute $5y$ to each term: $5y(2y^2) + 5y(-3y) + 5y(1) = 10y^3 - 15y^2 + 5y$.
• To find the product of $(a+6)(a^2-2a+3)$, distribute $a$ and $6$: $a(a^2-2a+3) + 6(a^2-2a+3) = a^3-2a^2+3a + 6a^2-12a+18$. Combining like terms gives $a^3 + 4a^2 - 9a + 18$.

Explanation

Multiplying polynomials means every term in the first expression must be multiplied by every term in the second. The distributive property ensures no term is missed. After multiplying, you clean up by combining any like terms.

Property

FOIL is a shortcut for multiplying two binomials. It stands for First, Outer, Inner, Last. Given two binomials $(ax+b)$ and $(cx+d)$, the product is found by summing the products of the First terms ($acx^2$), Outer terms ($adx$), Inner terms ($bcx$), and Last terms ($bd$).

Examples

• To find $(3x+4)(2x-1)$ using FOIL: (First: $6x^2$) + (Outer: $-3x$) + (Inner: $8x$) + (Last: $-4$). Combining terms gives $6x^2 + 5x - 4$.
• To find $(y-5)(y-3)$ using FOIL: (First: $y^2$) + (Outer: $-3y$) + (Inner: $-5y$) + (Last: $15$). Combining terms gives $y^2 - 8y + 15$.
• To find $(5a+2)(a+6)$ using FOIL: (First: $5a^2$) + (Outer: $30a$) + (Inner: $2a$) + (Last: $12$). Combining terms gives $5a^2 + 32a + 12$.

Explanation

FOIL is a memory aid for the distributive property when multiplying two binomials. It organizes the four necessary multiplications so you can quickly find the product and then combine the middle terms.

Property

When a binomial is squared, the result is a perfect square trinomial. The formula is:

Examples

• To expand $(2x+5)^2$, use the pattern: $(2x)^2 + 2(2x)(5) + 5^2 = 4x^2 + 20x + 25$.
• To expand $(y-4)^2$, use the pattern: $y^2 + 2(y)(-4) + (-4)^2 = y^2 - 8y + 16$.
• To expand $(3a-2b)^2$, use the pattern: $(3a)^2 + 2(3a)(-2b) + (-2b)^2 = 9a^2 - 12ab + 4b^2$.

Explanation

This is a shortcut for squaring a binomial. Instead of using FOIL, you can square the first term, square the last term, and multiply the two terms together and double the result for the middle term.

Property

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

Examples

• Multiply $(x + 8)(x - 8)$. The result is the square of the first term, $x^2$, minus the square of the second term, $8^2$. So, $x^2 - 64$.

• Multiply $(5y + 2)(5y - 2)$. This gives $(5y)^2 - 2^2$, which simplifies to $25y^2 - 4$.