Algebra is the main field of mathematics. 9^{th} 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 9^{th} grade algebra.

**Example 1: **Find the sum of x^{3}y + x^{2}y^{2 }– 3xy^{3 }and x^{3 }– 3x^{3}y + y^{3 }+ 4xy^{3}.

**Answer:**

(x^{3}y + x^{2}y^{2 }– 3xy^{3}) + (x^{3 }– 3x^{3}y + y^{3 }+ 4xy^{3})

= x^{3}y + x^{2}y^{2 }– 3xy^{3 }+ x^{3 }– 3x^{3}y + y^{3 }+ 4xy^{3}

= (x^{3}y – 3x^{3}y) + (x^{2}y^{2}) + (–3xy^{3 }+ 4xy^{3}) + (x^{3}) + (y^{3})

= –2x^{3}y + x^{2}y^{2 }+ xy^{3 }+ x^{3 }+ y^{3 }.

**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) =x^{3 }+ (a + b + c)x^{2 }+ (ab + bc + ca)x + abc

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

= x^{3 }+ (4 + 3 + 5)x^{2 }+ [4 × 3 + 3 × 5 + 5 × 4]x + 4 × 3 × 5

= x^{3 }+ 12x^{2 }+ [12 + 15 + 20] x + 60

= x^{3 }+ 12x^{2 }+ 47x + 60.

**Example 3: **Factorize 16x^{4}y^{2 }– 25.

**Answer:**

Since 16x^{4}y^{2 }= (4x^{2}y)^{2 }and 25 = (5)^{2}, we have

16x^{4}y^{2 }– 25 = (4x^{2}y)^{2 }– (5)^{2 }= (4x^{2}y + 5) (4x^{2}y – 5)

**Example 4: **If (x + p)(x + q) = x^{2 }– 5x – 300, find the value of p^{2 }+ q^{2}.

**Answer:**

By product formula, we have (x + p) (x + q) = x^{2 }+ (p + q)x + pq.

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

Now, we have p^{2 }+ q^{2 }= (p + q)^{2 }– 2 pq = (–5)^{2 }–2(–300) = 25 + 600 = 625.

**Example 5: **If a+b=2 and a^{2}+b^{2}=8, find a^{3}+b^{3 }and a^{4}+b^{4}.

**Answer:**

a^{2 }+ 2ab + b^{2}= (a+b)^{2 }

2ab = (a+b)^{2 }– (a^{2 }+ b^{2}) = (2)^{2 }– (8) = 4 – 8 = – 4.

∴ ab = 21(2ab) = 21(–4) = – 2.

a^{3}+ b^{3 }= (a+b)^{3 }– 3ab(a+b) = (2)^{3 }– 3(–2)(2) = 8 – 3 (– 4) = 8 + 12 = 20.

These are the main problems in 9^{th} grade algebra.