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Sagot :
Sure! Let's find the product for each pair of polynomials step by step:
### (a) [tex]\( (x^2 + 3x + 2) (x + 1) \)[/tex]
First, we expand the product:
[tex]\[ (x^2 + 3x + 2)(x + 1) = x^2(x + 1) + 3x(x + 1) + 2(x + 1) \][/tex]
Distribute each term:
[tex]\[ = x^3 + x^2 + 3x^2 + 3x + 2x + 2 \][/tex]
Combine like terms:
[tex]\[ = x^3 + 4x^2 + 5x + 2 \][/tex]
So, the product is [tex]\( x^3 + 4x^2 + 5x + 2 \)[/tex].
### (b) [tex]\( (x^2 - 7x + 12) (x - 3) \)[/tex]
First, we expand the product:
[tex]\[ (x^2 - 7x + 12)(x - 3) = x^2(x - 3) - 7x(x - 3) + 12(x - 3) \][/tex]
Distribute each term:
[tex]\[ = x^3 - 3x^2 - 7x^2 + 21x + 12x - 36 \][/tex]
Combine like terms:
[tex]\[ = x^3 - 10x^2 + 33x - 36 \][/tex]
So, the product is [tex]\( x^3 - 10x^2 + 33x - 36 \)[/tex].
### (c) [tex]\( (a^2 - 11a + 30) (a - 5) \)[/tex]
First, we expand the product:
[tex]\[ (a^2 - 11a + 30)(a - 5) = a^2(a - 5) - 11a(a - 5) + 30(a - 5) \][/tex]
Distribute each term:
[tex]\[ = a^3 - 5a^2 - 11a^2 + 55a + 30a - 150 \][/tex]
Combine like terms:
[tex]\[ = a^3 - 16a^2 + 85a - 150 \][/tex]
So, the product is [tex]\( a^3 - 16a^2 + 85a - 150 \)[/tex].
### (d) [tex]\( (a^2 - 49a + 60) (2a + 5) \)[/tex]
First, we expand the product:
[tex]\[ (a^2 - 49a + 60)(2a + 5) = a^2(2a + 5) - 49a(2a + 5) + 60(2a + 5) \][/tex]
Distribute each term:
[tex]\[ = 2a^3 + 5a^2 - 98a^2 - 245a + 120a + 300 \][/tex]
Combine like terms:
[tex]\[ = 2a^3 - 93a^2 - 125a + 300 \][/tex]
So, the product is [tex]\( 2a^3 - 93a^2 - 125a + 300 \)[/tex].
### (e) [tex]\( (3x^2 + 10x + 3) (x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (3x^2 + 10x + 3)(x + 3) = 3x^2(x + 3) + 10x(x + 3) + 3(x + 3) \][/tex]
Distribute each term:
[tex]\[ = 3x^3 + 9x^2 + 10x^2 + 30x + 3x + 9 \][/tex]
Combine like terms:
[tex]\[ = 3x^3 + 19x^2 + 33x + 9 \][/tex]
So, the product is [tex]\( 3x^3 + 19x^2 + 33x + 9 \)[/tex].
### (f) [tex]\( (2x^2 + 11x + 5) (2x + 1) \)[/tex]
First, we expand the product:
[tex]\[ (2x^2 + 11x + 5)(2x + 1) = 2x^2(2x + 1) + 11x(2x + 1) + 5(2x + 1) \][/tex]
Distribute each term:
[tex]\[ = 4x^3 + 2x^2 + 22x^2 + 11x + 10x + 5 \][/tex]
Combine like terms:
[tex]\[ = 4x^3 + 24x^2 + 21x + 5 \][/tex]
So, the product is [tex]\( 4x^3 + 24x^2 + 21x + 5 \)[/tex].
### (g) [tex]\( (5x^2 + 11x + 2) (4x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (5x^2 + 11x + 2)(4x + 3) = 5x^2(4x + 3) + 11x(4x + 3) + 2(4x + 3) \][/tex]
Distribute each term:
[tex]\[ = 20x^3 + 15x^2 + 44x^2 + 33x + 8x + 6 \][/tex]
Combine like terms:
[tex]\[ = 20x^3 + 59x^2 + 41x + 6 \][/tex]
So, the product is [tex]\( 20x^3 + 59x^2 + 41x + 6 \)[/tex].
### (h) [tex]\( (2x^2 + 17x + 21) (2x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (2x^2 + 17x + 21)(2x + 3) = 2x^2(2x + 3) + 17x(2x + 3) + 21(2x + 3) \][/tex]
Distribute each term:
[tex]\[ = 4x^3 + 6x^2 + 34x^2 + 51x + 42x + 63 \][/tex]
Combine like terms:
[tex]\[ = 4x^3 + 40x^2 + 93x + 63 \][/tex]
So, the product is [tex]\( 4x^3 + 40x^2 + 93x + 63 \)[/tex].
To summarize, the products are:
(a) [tex]\( x^3 + 4x^2 + 5x + 2 \)[/tex]
(b) [tex]\( x^3 - 10x^2 + 33x - 36 \)[/tex]
(c) [tex]\( a^3 - 16a^2 + 85a - 150 \)[/tex]
(d) [tex]\( 2a^3 - 93a^2 - 125a + 300 \)[/tex]
(e) [tex]\( 3x^3 + 19x^2 + 33x + 9 \)[/tex]
(f) [tex]\( 4x^3 + 24x^2 + 21x + 5 \)[/tex]
(g) [tex]\( 20x^3 + 59x^2 + 41x + 6 \)[/tex]
(h) [tex]\( 4x^3 + 40x^2 + 93x + 63 \)[/tex]
### (a) [tex]\( (x^2 + 3x + 2) (x + 1) \)[/tex]
First, we expand the product:
[tex]\[ (x^2 + 3x + 2)(x + 1) = x^2(x + 1) + 3x(x + 1) + 2(x + 1) \][/tex]
Distribute each term:
[tex]\[ = x^3 + x^2 + 3x^2 + 3x + 2x + 2 \][/tex]
Combine like terms:
[tex]\[ = x^3 + 4x^2 + 5x + 2 \][/tex]
So, the product is [tex]\( x^3 + 4x^2 + 5x + 2 \)[/tex].
### (b) [tex]\( (x^2 - 7x + 12) (x - 3) \)[/tex]
First, we expand the product:
[tex]\[ (x^2 - 7x + 12)(x - 3) = x^2(x - 3) - 7x(x - 3) + 12(x - 3) \][/tex]
Distribute each term:
[tex]\[ = x^3 - 3x^2 - 7x^2 + 21x + 12x - 36 \][/tex]
Combine like terms:
[tex]\[ = x^3 - 10x^2 + 33x - 36 \][/tex]
So, the product is [tex]\( x^3 - 10x^2 + 33x - 36 \)[/tex].
### (c) [tex]\( (a^2 - 11a + 30) (a - 5) \)[/tex]
First, we expand the product:
[tex]\[ (a^2 - 11a + 30)(a - 5) = a^2(a - 5) - 11a(a - 5) + 30(a - 5) \][/tex]
Distribute each term:
[tex]\[ = a^3 - 5a^2 - 11a^2 + 55a + 30a - 150 \][/tex]
Combine like terms:
[tex]\[ = a^3 - 16a^2 + 85a - 150 \][/tex]
So, the product is [tex]\( a^3 - 16a^2 + 85a - 150 \)[/tex].
### (d) [tex]\( (a^2 - 49a + 60) (2a + 5) \)[/tex]
First, we expand the product:
[tex]\[ (a^2 - 49a + 60)(2a + 5) = a^2(2a + 5) - 49a(2a + 5) + 60(2a + 5) \][/tex]
Distribute each term:
[tex]\[ = 2a^3 + 5a^2 - 98a^2 - 245a + 120a + 300 \][/tex]
Combine like terms:
[tex]\[ = 2a^3 - 93a^2 - 125a + 300 \][/tex]
So, the product is [tex]\( 2a^3 - 93a^2 - 125a + 300 \)[/tex].
### (e) [tex]\( (3x^2 + 10x + 3) (x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (3x^2 + 10x + 3)(x + 3) = 3x^2(x + 3) + 10x(x + 3) + 3(x + 3) \][/tex]
Distribute each term:
[tex]\[ = 3x^3 + 9x^2 + 10x^2 + 30x + 3x + 9 \][/tex]
Combine like terms:
[tex]\[ = 3x^3 + 19x^2 + 33x + 9 \][/tex]
So, the product is [tex]\( 3x^3 + 19x^2 + 33x + 9 \)[/tex].
### (f) [tex]\( (2x^2 + 11x + 5) (2x + 1) \)[/tex]
First, we expand the product:
[tex]\[ (2x^2 + 11x + 5)(2x + 1) = 2x^2(2x + 1) + 11x(2x + 1) + 5(2x + 1) \][/tex]
Distribute each term:
[tex]\[ = 4x^3 + 2x^2 + 22x^2 + 11x + 10x + 5 \][/tex]
Combine like terms:
[tex]\[ = 4x^3 + 24x^2 + 21x + 5 \][/tex]
So, the product is [tex]\( 4x^3 + 24x^2 + 21x + 5 \)[/tex].
### (g) [tex]\( (5x^2 + 11x + 2) (4x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (5x^2 + 11x + 2)(4x + 3) = 5x^2(4x + 3) + 11x(4x + 3) + 2(4x + 3) \][/tex]
Distribute each term:
[tex]\[ = 20x^3 + 15x^2 + 44x^2 + 33x + 8x + 6 \][/tex]
Combine like terms:
[tex]\[ = 20x^3 + 59x^2 + 41x + 6 \][/tex]
So, the product is [tex]\( 20x^3 + 59x^2 + 41x + 6 \)[/tex].
### (h) [tex]\( (2x^2 + 17x + 21) (2x + 3) \)[/tex]
First, we expand the product:
[tex]\[ (2x^2 + 17x + 21)(2x + 3) = 2x^2(2x + 3) + 17x(2x + 3) + 21(2x + 3) \][/tex]
Distribute each term:
[tex]\[ = 4x^3 + 6x^2 + 34x^2 + 51x + 42x + 63 \][/tex]
Combine like terms:
[tex]\[ = 4x^3 + 40x^2 + 93x + 63 \][/tex]
So, the product is [tex]\( 4x^3 + 40x^2 + 93x + 63 \)[/tex].
To summarize, the products are:
(a) [tex]\( x^3 + 4x^2 + 5x + 2 \)[/tex]
(b) [tex]\( x^3 - 10x^2 + 33x - 36 \)[/tex]
(c) [tex]\( a^3 - 16a^2 + 85a - 150 \)[/tex]
(d) [tex]\( 2a^3 - 93a^2 - 125a + 300 \)[/tex]
(e) [tex]\( 3x^3 + 19x^2 + 33x + 9 \)[/tex]
(f) [tex]\( 4x^3 + 24x^2 + 21x + 5 \)[/tex]
(g) [tex]\( 20x^3 + 59x^2 + 41x + 6 \)[/tex]
(h) [tex]\( 4x^3 + 40x^2 + 93x + 63 \)[/tex]
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