Ninth Grade Mathematics: Addition, Subtraction & Multiplication

Arithmetic operations on polynomials follow the fundamental laws of algebra that govern the real numbers. When polynomials are added, subtracted, or multiplied, the results are always new polynomials, demonstrating that the set of polynomials is closed under these three operations.

Addition of Polynomials

Adding polynomials involves combining like terms—terms that contain the exact same variables raised to the exact same exponents. To add two polynomials, group like terms together and add their numerical coefficients while leaving the variable parts unchanged.

Addition Example:
Add P(x) = 4*x^3 - 5*x^2 + 3*x - 7  and  Q(x) = 2*x^3 + 7*x^2 - 8*x + 12

Horizontal Method:
(4*x^3 - 5*x^2 + 3*x - 7) + (2*x^3 + 7*x^2 - 8*x + 12)
= (4 + 2)*x^3 + (-5 + 7)*x^2 + (3 - 8)*x + (-7 + 12)
= 6*x^3 + 2*x^2 - 5*x + 5

Vertical (Column) Method:
    4*x^3 - 5*x^2 + 3*x -  7
+   2*x^3 + 7*x^2 - 8*x + 12
----------------------------
    6*x^3 + 2*x^2 - 5*x +  5

The degree of the sum of two polynomials is at most equal to the maximum of the degrees of the two individual polynomials. It can be strictly less if the leading terms have opposite coefficients and cancel out.

Subtraction of Polynomials

Subtracting a polynomial is equivalent to adding its additive inverse. To subtract Q(x) from P(x), distribute a negative sign to every term of Q(x) (which flips the sign of every coefficient), and then combine like terms.

Subtraction Example:
Subtract Q(x) = 3*x^3 - 4*x^2 + 5  from  P(x) = 5*x^3 + 2*x^2 - 6*x + 1

P(x) - Q(x) = (5*x^3 + 2*x^2 - 6*x + 1) - (3*x^3 - 4*x^2 + 5)
            = 5*x^3 + 2*x^2 - 6*x + 1 - 3*x^3 + 4*x^2 - 5
            = (5 - 3)*x^3 + (2 + 4)*x^2 - 6*x + (1 - 5)
            = 2*x^3 + 6*x^2 - 6*x - 4

A common student error in polynomial subtraction is failing to distribute the negative sign to all terms inside the second set of parentheses. Be vigilant in changing every sign before combining.

Multiplication of Polynomials

Multiplication of polynomials relies entirely on the distributive property of multiplication over addition, along with the Product Law of exponents: a^m * a^n = a^(m + n). Every term of the first polynomial must be multiplied by every term of the second polynomial.

The Distributive Law:
(A + B) * (C + D + E) = A*C + A*D + A*E + B*C + B*D + B*E

Multiplying a Binomial by a Trinomial

To multiply a binomial by a trinomial, execute six distinct multiplications and collect like terms systematically.

Multiply: (2*x - 3) * (4*x^2 + 5*x - 6)

Step 1: Distribute 2*x across the trinomial:
        2*x * (4*x^2) = 8*x^3
        2*x * (5*x)   = 10*x^2
        2*x * (-6)    = -12*x

Step 2: Distribute -3 across the trinomial:
        -3 * (4*x^2)  = -12*x^2
        -3 * (5*x)    = -15*x
        -3 * (-6)     = 18

Step 3: Combine all six terms:
        8*x^3 + (10*x^2 - 12*x^2) + (-12*x - 15*x) + 18
        = 8*x^3 - 2*x^2 - 27*x + 18

Degree of a Product

Unlike addition where degrees do not add, the degree of the product of two non-zero polynomials is always strictly equal to the sum of their individual degrees:

deg( P(x) * Q(x) ) = deg( P(x) ) + deg( Q(x) )

For example, multiplying a degree-2 quadratic by a degree-3 cubic will always produce a degree-5 quintic polynomial.

Chapter Exercises

Exercise 1: Compute the sum and difference P(x) + Q(x) and P(x) - Q(x) for P(x) = 7x^4 - 3x^2 + 5x - 8 and Q(x) = 2x^4 + 4x^3 - 3x^2 - 9*x + 4.

Exercise 2: Multiply the polynomial (3x^2 - 2x + 4) by the binomial (2*x - 5) and write the resulting polynomial in standard form.

Exercise 3: If P(x) is a polynomial of degree 4 and Q(x) is a polynomial of degree 3, determine the maximum possible degree of P(x) + Q(x), the minimum possible degree of P(x) + Q(x), and the exact degree of P(x) * Q(x).

Solutions and Explanations

Solution 1: To find P(x) + Q(x), group like terms: (7 + 2)x^4 + (0 + 4)x^3 + (-3 - 3)x^2 + (5 - 9)x + (-8 + 4) = 9x^4 + 4x^3 - 6x^2 - 4x - 4. To find P(x) - Q(x), distribute the negative sign to every term of Q(x) and add: (7 - 2)x^4 + (0 - 4)x^3 + (-3 - (-3))x^2 + (5 - (-9))x + (-8 - 4) = 5x^4 - 4x^3 + 0x^2 + 14x - 12 = 5x^4 - 4x^3 + 14*x - 12.

Solution 2: Multiply each term of (3x^2 - 2x + 4) by each term of (2x - 5). Distributing 2x yields: 2x * (3x^2) = 6x^3; 2x * (-2x) = -4x^2; 2x * (4) = 8x. Distributing -5 yields: -5 * (3x^2) = -15x^2; -5 * (-2x) = 10x; -5 * (4) = -20. Summing these terms and combining like powers gives: 6x^3 + (-4x^2 - 15x^2) + (8x + 10x) - 20 = 6x^3 - 19x^2 + 18x - 20.

Solution 3: When adding two polynomials of different degrees, the leading term of the higher-degree polynomial cannot be cancelled by any term from the lower-degree polynomial. Because deg(P) = 4 and deg(Q) = 3, the x^4 term of P(x) remains unaffected by addition, meaning the degree of P(x) + Q(x) must be exactly 4 (both the maximum and minimum possible degree are 4). For the product P(x) * Q(x), the degree of a product is the sum of the degrees of the factors: deg(P * Q) = deg(P) + deg(Q) = 4 + 3 = 7.