Polynomial Identities

Sometimes, you come across polynomial problems that defy rationality. They just look at you, challenging you to think of a focus. There seems to be no intuitive way to solve the problem. In many of these situations, a polynomial identity has been created to slam the door open and ease the problem.

All identities of polynomials are essentially just true equations, but they are very helpful for demonstrating the connection between two seemingly unrelated expressions. Since the identity allows you to make that connection, you can use it to convert back and forth between the two expressions.

Some Useful Identities

There are many popular polynomial identities in the world of Mathematics, and here are some valuable ones:

\[(a + b)^2 = a^2 + 2 ab + b^2\]

This can speed up your FOIL (First – Outer – Inner – Last) factoring and multiplication. When a binomial is squared, it always breaks down into the same expression. A similar identity is one in which the terms are subtracted:

\[(a – b)^2 = a^2– 2 ab + b^2\]

When you see a polynomial in any form on the right-hand side of the identity, you know that it will be included in the expression on the left. Remember, these are not the same as \[a^2 + b^2\], which doesn't have an identity to factor.

The identity for cubes of the sum of two variables is \[(a+b)^3 = a^3+3a^2b+3ab^2+b^3\]

The identity for cubes of difference of two variables is \[(a-b)^3 = a^3-3a^2b+3ab^2-b^3\].

Difference Between Squares

The difference between the square’s identity can save you from factoring and multiplication:

\[a^2 – b^2 = (a + b) (a – b)\]

This identity is so useful that you will find yourself looking through your polynomial problems, delighted when you see any form of difference between squares. Again, the sum of squares \[a ^2 + b ^2\] does not factor (at least not in real numbers) and does not have a useful identity like this.

Sum/Difference of Cubes

The lifesavers in terms of polynomial factorization are the sum of cubes and the difference of cube identities. These are extremely helpful to remember.

\[a ^3 + b ^3 = (a + b) (a^2 – ab + b^2)\] is the sum of cubes and \[a^3 – b^3 = (a – b) (a^2 + ab + b^2)\] is the difference of cubes.

Quadratic formula

Although it's not as fun trying to remember, it is one of its most useful polynomial identities. Allows you to divide any quadratic expression (polynomial with a squared term as its highest order expression) into its factors:

If $ax^2 + bx + c = 0,$ then you can use the formula we just covered to find the factors (real or imaginary). Note that a, b, and c represent the coefficients (as in the numerical part of the terms) that appear in the quadratic equation. When you use this identity to solve a quadratic equation, you may end up with one or two real values ​​for x, or you may try to take the square root of a negative number, which means you end up with imaginary results!

\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Here are a couple of common polynomial identities that may be helpful while wandering through algebraic minefields, especially when working with quadratic equations:

(a + b) (c + d) = ac + ad + bc + bd (the FOIL formula)

\[x ^2+ (a + b) x + ab = (x + a) (x + b)\] (another FOIL formula)

Now let’s see some polynomial identity examples.

Solved Examples

Example 1: Solve the expression \[(x+2)^2\].

Sol: Put a=x and b=2 in $(a + b)^2 = a^2 + 2 ab + b^2$

\[x^2+2(x)(2) +2^2\]

\[= x^2+4x+4\]

Example 2: Solve the expression (6+4) (6-4).

Sol: Put a=6 and b=4 in polynomial identity \[( a + b ) ( a – b )=a^2 – b^2\]

\[(6+4) (6-4) = 6^2 - 4^2\]

=36-16

=20

Example 3: Solve the expression \[x^2+6x+2\] using quadratic formula.

Sol: Quadratic formula is \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Compare \[a^2x+bx+c\] with \[x^2+6x+2\]:

Here, a=1, b=6, c=2

So,

\[x = \dfrac{{ - 6 \pm \sqrt {{6^2} - 4(1)(2)} }}{{2(1)}}\]

\[x = \dfrac{{ - 6 \pm \sqrt {36 - 8} }}{2}\]

\[x = \dfrac{{ - 6 \pm \sqrt {28} }}{2}\]

\[x = \dfrac{{ - 6 \pm 2\sqrt 7 }}{2}\]

\[x = - 3 \pm \sqrt 7 \]

Conclusion

True equations with broad applications in your algebra studies are important polynomial identities. Your Maths homework will be simpler the more comfortable you are with these, like:

• \[(a + b)^2 = a^2 + 2 ab + b^2\], or \[(a – b)^2= a^2 – 2 ab + b^2\]

• The difference between the identity of squares: \[a^2– b^2 = (a + b) (a – b)\]

• The sum of the identity of the cubes: \[a^3+ b ^3 = (a + b) (a^2 – ab + b ^2)\]

• The identity difference of the cubes: \[a ^3– b^3 = (a – b) (a^2 + ab + b^2)\]

• The FOIL formula: (a + b) (c + d) = ac + ad + bc + bd

• Another FOIL formula: \[x^2 + (a + b) x + ab = (x + a) (x + b)\]

• The quadratic formula, which allows you to decompose any quadratic expression into its factors.

Practice Questions

Q 1: Solve the expression using polynomial identities \[(6+4)^2\].

Ans: 100

Q 2: Solve the expression \[(100^2-4^2)\].

Sol: 9984