MULTIPLICATION OF ALGEBRAIC EXPRESSION OF TRINOMIAL BY TRINOMIAL -
Multiplying two trinomials together involves applying the distributive property multiple times. Let's say you have two trinomials,
A(x) = ax² + bx + c, and
B(x) = dx²+ ex +f, and you want to multiply them:
A(x) ⋅ B(x) = (ax²+ bx + c) (dx²+ ex + f)
To multiply these two trinomials together, you can use the distributive property repeatedly for each term in the first trinomial:
1. Multiply the first term of A(x) (ax²) by each term in B(x):-
(ax²) (dx²) + (ax²) (ex) + (ax²) (f)
2. Multiply the second term of A(x) (bx) by each term in B(x):-
(bx) (dx²) + (bx) (ex) + (bx) (f)
3. Multiply the third term of A(x) (c) by each term in B(x):-
(c) (dx²) + (c) (ex) + (c) (f)
Now, you have several terms, and you can combine like terms by adding or subtracting them:-
1. Combine like terms in the first set:-
2(ax²) (dx²) + (ax²) (ex) + (ax²) (f) = dx⁴ + aex³+ afx²
2. Combine like terms in the second set:-
(bx) (dx²) + (bx) (ex) + (bx) (f) = bdx³ + bex² + bfx
3. Combine like terms in the third set:-
(c) (dx²) + (c) (ex) + (c) (f) = cdx²+ cex + cf
Now, you can put all these together to get the final expression:-
A(x) ⋅ B(x) = (adx⁴ + aex³+ afx²) + (bdx³+ bex²+ bfx) + (cdx²+ cex + cf)
This is the result of multiplying two trinomials together. You can further simplify or factor the expression if needed.
Another Way Of Understanding -
Multiplying two algebraic expressions, each of which is a trinomial (a polynomial with three terms), can be done using the distributive property multiple times. Here's how you can multiply two trinomials step by step:
Let's say you have two trinomials, such as:-
A(x) = ax²+ bx + c
B(x) = dx²+ ex + f
To multiply these two trinomials, you can use the distributive property multiple times. Multiply each term in the first trinomial by each term in the second trinomial and then combine like terms.
1. Multiply the first term of the first trinomial (2ax²) by each term in the second trinomial:-
ax²(dx²) + ax²(ex) + ax²(f)
2. Multiply the second term of the first trinomial (bx) by each term in the second trinomial:-
bx (dx²) + bx (ex) + bx (f)
3. Multiply the third term of the first trinomial (c) by each term in the second trinomial:-
c (dx²) + c (ex) + c (f)
4. Now, you have several terms, each of which is a product of a term from the first trinomial and a term from the second trinomial.
5. Combine like terms by adding or subtracting them. Group terms with the same power of x:-
(ad) x⁴ + (ae + bd) x³+ (af + be + cd) x² + (bf + ce) x + cf
This is the result of multiplying the two trinomials A(x) and B(x).
The final expression is another trinomial, where each term is the result of multiplying corresponding terms from the original trinomials and then combining like terms.
Example.1) Multiply the trinomial A(x) = 2x²+ 3x − 1 by the trinomial B(x) = x²− 2x + 4.
Ans.)
Step 1:- Multiply the first term of A(x) by each term in B(x):
2x². (x²) + 2x² . (−2x) + 2x² . (4)
Step 2:- Multiply the second term of A(x) by each term in B(x):
3x . (x²) + 3x . (−2x) + 3x . (4)
Step 3:- Multiply the third term of A(x) by each term in B(x):
− 1(x²) − 1(−2x) − 1(4)
Step 4:- Simplify each of these products:
2x⁴ − 4x³+ 8x²+ 3x³− 6x²+ 12x − x² + 2x − 4
Step 5:- Combine like terms:
2x⁴ − x³+ x²+ 14x − 4
So, the result of multiplying A(x) and B(x) is the trinomial:-
2x⁴ − x³ + x² + 14x − 4
This is the product of the two given trinomials. (Ans.)
Example.2) Multiplying two trinomials:
A(x) = 2x²+ 3x − 4
B(x) = 4x²− x + 5
Ans.)
Now, we want to multiply A(x) by B(x). We'll use the steps mentioned earlier:
1. Multiply the first term of A(x) by each term in B(x):-
2x². (4x²) + 2x². (−x) + 2x². (5)
2. Multiply the second term of A(x) by each term in B(x):-
3x . (4x²) + 3x .(−x) + 3x . (5)
3. Multiply the third term of A(x) by each term in B(x):-
− 4 . (4x²) − 4 . (−x) − 4 . (5)
4. Now, let's compute each of these products:
First set of terms:- 8x⁴ − 2x³ + 10x²
Second set of terms:- 12x³− 3x² + 15x
Third set of terms:− 16x² + 4x − 20
5. Finally, combine like terms by adding or subtracting them:-
(8x⁴ − 2x³+ 10x²) + (12x³− 3x²+ 15x) + (−16x²+ 4x − 20)
6. Simplify further if needed:-
8x⁴ + 10x³− 2x3 + 12x² − 3x² − 16x²+ 15x + 4x −20
7. Combine like terms once more:-
8x⁴ + 8x³− 7x² + 19x − 20
So, the result of multiplying the two trinomials A(x) and B(x) is:-
8x⁴ + 8x³− 7x² + 19x − 20 (Ans.)