Online Algebra Help

Answers for 9th Grade Algebra

Algebra is the main field of mathematics. 9th grade algebra mainly contains polynomials, factorization, simplifying algebraic identities and expressions. Basically we can use letters like a, b, x and y to denote numbers. Using the basic operations add,subtract, product , divide or extraction of roots on these symbols and real numbers, we obtain what are called algebraic expressions. This contains following types of problems like Polynomials, Algebraic Identities, factorization and Division of a Polynomial by a Polynomial. These are the main topics in 9th grade algebra.



9th grade Algebra Problems with answers


Example 1: Find the sum of x3y + x2y2 – 3xy3 and x3 – 3x3y + y3 + 4xy3.

Answer:

(x3y + x2y2 – 3xy3) + (x3 – 3x3y + y3 + 4xy3)

= x3y + x2y2 – 3xy3 + x3 – 3x3y + y3 + 4xy3

= (x3y – 3x3y) + (x2y2) + (–3xy3 + 4xy3) + (x3) + (y3)

= –2x3y + x2y2 + xy3 + x3 + y3 .

Example 2: Find the expansions of (x + 4) (x + 3) (x + 5)

Answer:

 (x + 4)(x + 3)(x + 5)

Here we can use the following formula.

(x + a)(x + b)(x + c) =x3 + (a + b + c)x2 + (ab + bc + ca)x + abc

Here a="4" and b="3" and c="5

= x3 + (4 + 3 + 5)x2 + [4 × 3 + 3 × 5 + 5 × 4]x + 4 × 3 × 5

= x3 + 12x2 + [12 + 15 + 20] x + 60

= x3 + 12x2 + 47x + 60.


Example 3: Factorize 16x4y2 – 25.

Answer:

Since 16x4y2 = (4x2y)2 and 25 = (5)2, we have

16x4y2 – 25 = (4x2y)2 – (5)2 = (4x2y + 5) (4x2y – 5)


Example 4: If (x + p)(x + q) = x2 – 5x – 300, find the value of p2 + q2.

Answer:

By product formula, we have (x + p) (x + q) = x2 + (p + q)x + pq.

So, by comparison, we get p + q = –5, pq = –300.

Now, we have p2 + q2 = (p + q)2 – 2 pq = (–5)2 –2(–300) = 25 + 600 = 625.


Example 5: If a+b=2 and a2+b2=8, find a3+b3 and a4+b4.

Answer:

a2 + 2ab + b2= (a+b)2

2ab = (a+b)2 – (a2 + b2) = (2)2 – (8) = 4 – 8 = – 4.

∴ ab = 21(2ab) = 21(–4) = – 2.

a3+ b3 = (a+b)3 – 3ab(a+b) = (2)3 – 3(–2)(2) = 8 – 3 (– 4) = 8 + 12 = 20.

These are the main problems in 9th grade algebra.