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Let's walk through the process of factorizing each of the given polynomials step-by-step:
### (a) \( a^2 + 12a + 36 \)
1. Look for two numbers that multiply to \( 36 \) (the constant term) and add to \( 12 \) (the coefficient of the linear term \( a \)).
2. The numbers \( 6 \) and \( 6 \) fit these conditions because:
[tex]\[ 6 \times 6 = 36 \][/tex]
[tex]\[ 6 + 6 = 12 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ a^2 + 12a + 36 = (a + 6)(a + 6) \][/tex]
4. Simplify the factors:
[tex]\[ (a + 6)^2 \][/tex]
### (b) \( y^2 + 14y + 49 \)
1. Look for two numbers that multiply to \( 49 \) (the constant term) and add to \( 14 \) (the coefficient of the linear term \( y \)).
2. The numbers \( 7 \) and \( 7 \) fit these conditions because:
[tex]\[ 7 \times 7 = 49 \][/tex]
[tex]\[ 7 + 7 = 14 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ y^2 + 14y + 49 = (y + 7)(y + 7) \][/tex]
4. Simplify the factors:
[tex]\[ (y + 7)^2 \][/tex]
### (c) \( p^2 + 22p + 121 \)
1. Look for two numbers that multiply to \( 121 \) (the constant term) and add to \( 22 \) (the coefficient of the linear term \( p \)).
2. The numbers \( 11 \) and \( 11 \) fit these conditions because:
[tex]\[ 11 \times 11 = 121 \][/tex]
[tex]\[ 11 + 11 = 22 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ p^2 + 22p + 121 = (p + 11)(p + 11) \][/tex]
4. Simplify the factors:
[tex]\[ (p + 11)^2 \][/tex]
### (d) \( 4a^2 + 20a + 25 \)
1. Notice that the leading coefficient of the quadratic term is \( 4 \), and the constant term is \( 25 \).
2. Look for two numbers that multiply to \( 4 \times 25 = 100 \) and add to \( 20 \) (linear term coefficient).
3. The numbers \( 10 \) and \( 10 \) fit these conditions because:
[tex]\[ 10 \times 10 = 100 \][/tex]
[tex]\[ 10 + 10 = 20 \][/tex]
4. Rewrite the factors in terms of \( a \):
[tex]\[ 4a^2 + 20a + 25 = (2a + 5)(2a + 5) \][/tex]
5. Simplify the factors:
[tex]\[ (2a + 5)^2 \][/tex]
### (e) \( 9r^2 + 60r + 100 \)
1. Factor out the quadratic and constant terms.
2. Look for two numbers that multiply to \( 900 \) (i.e., \( 9 \times 100 \)) and add to \( 60 \) (linear term coefficient).
3. The numbers \( 30 \) and \( 30 \) fit these conditions because:
[tex]\[ 30 \times 30 = 900 \][/tex]
[tex]\[ 30 + 30 = 60 \][/tex]
4. Rewrite the factors in terms of \( r \):
[tex]\[ 9r^2 + 60r + 100 = (3r + 10)(3r + 10) \][/tex]
5. Simplify the factors:
[tex]\[ (3r + 10)^2 \][/tex]
### (f) \( 36x^2 + 84x + 49 \)
1. Factor out the quadratic and constant terms.
2. Look for two numbers that multiply to \( 1764 \) (i.e., \( 36 \times 49 \)) and add to \( 84 \) (linear term coefficient).
3. The numbers \( 42 \) and \( 42 \) fit these conditions because:
[tex]\[ 42 \times 42 = 1764 \][/tex]
[tex]\[ 42 + 42 = 84 \][/tex]
4. Rewrite the factors in terms of \( x \):
[tex]\[ 36x^2 + 84x + 49 = (6x + 7)(6x + 7) \][/tex]
5. Simplify the factors:
[tex]\[ (6x + 7)^2 \][/tex]
### (g) \( x^2 - 8x + 16 \)
1. Look for two numbers that multiply to \( 16 \) (the constant term) and add to \( -8 \) (the coefficient of the linear term \( x \)).
2. The numbers \( -4 \) and \( -4 \) fit these conditions because:
[tex]\[ -4 \times -4 = 16 \][/tex]
[tex]\[ -4 + -4 = -8 \][/tex]
3. Rewrite the factors:
[tex]\[ x^2 - 8x + 16 = (x - 4)(x - 4) \][/tex]
4. Simplify the factors:
[tex]\[ (x - 4)^2 \][/tex]
### (h) \( a^2 - 18a + 81 \)
1. Look for two numbers that multiply to \( 81 \) (the constant term) and add to \( -18 \) (the coefficient of the linear term \( a \)).
2. The numbers \( -9 \) and \( -9 \) fit these conditions because:
[tex]\[ -9 \times -9 = 81 \][/tex]
[tex]\[ -9 + -9 = -18 \][/tex]
3. Rewrite the factors:
[tex]\[ a^2 - 18a + 81 = (a - 9)(a - 9) \][/tex]
4. Simplify the factors:
[tex]\[ (a - 9)^2 \][/tex]
### (i) \( p^2 - 26p + 169 \)
1. Look for two numbers that multiply to \( 169 \) (the constant term) and add to \( -26 \) (the coefficient of the linear term \( p \)).
2. The numbers \( -13 \) and \( -13 \) fit these conditions because:
[tex]\[ -13 \times -13 = 169 \][/tex]
[tex]\[ -13 + -13 = -26 \][/tex]
3. Rewrite the factors:
[tex]\[ p^2 - 26p + 169 = (p - 13)(p - 13) \][/tex]
4. Simplify the factors:
[tex]\[ (p - 13)^2 \][/tex]
These factorized forms of the given polynomials are:
- (a) \( (a + 6)^2 \)
- (b) \( (y + 7)^2 \)
- (c) \( (p + 11)^2 \)
- (d) \( (2a + 5)^2 \)
- (e) \( (3r + 10)^2 \)
- (f) \( (6x + 7)^2 \)
- (g) \( (x - 4)^2 \)
- (h) \( (a - 9)^2 \)
- (i) [tex]\( (p - 13)^2 \)[/tex]
### (a) \( a^2 + 12a + 36 \)
1. Look for two numbers that multiply to \( 36 \) (the constant term) and add to \( 12 \) (the coefficient of the linear term \( a \)).
2. The numbers \( 6 \) and \( 6 \) fit these conditions because:
[tex]\[ 6 \times 6 = 36 \][/tex]
[tex]\[ 6 + 6 = 12 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ a^2 + 12a + 36 = (a + 6)(a + 6) \][/tex]
4. Simplify the factors:
[tex]\[ (a + 6)^2 \][/tex]
### (b) \( y^2 + 14y + 49 \)
1. Look for two numbers that multiply to \( 49 \) (the constant term) and add to \( 14 \) (the coefficient of the linear term \( y \)).
2. The numbers \( 7 \) and \( 7 \) fit these conditions because:
[tex]\[ 7 \times 7 = 49 \][/tex]
[tex]\[ 7 + 7 = 14 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ y^2 + 14y + 49 = (y + 7)(y + 7) \][/tex]
4. Simplify the factors:
[tex]\[ (y + 7)^2 \][/tex]
### (c) \( p^2 + 22p + 121 \)
1. Look for two numbers that multiply to \( 121 \) (the constant term) and add to \( 22 \) (the coefficient of the linear term \( p \)).
2. The numbers \( 11 \) and \( 11 \) fit these conditions because:
[tex]\[ 11 \times 11 = 121 \][/tex]
[tex]\[ 11 + 11 = 22 \][/tex]
3. Rewrite the quadratic expression:
[tex]\[ p^2 + 22p + 121 = (p + 11)(p + 11) \][/tex]
4. Simplify the factors:
[tex]\[ (p + 11)^2 \][/tex]
### (d) \( 4a^2 + 20a + 25 \)
1. Notice that the leading coefficient of the quadratic term is \( 4 \), and the constant term is \( 25 \).
2. Look for two numbers that multiply to \( 4 \times 25 = 100 \) and add to \( 20 \) (linear term coefficient).
3. The numbers \( 10 \) and \( 10 \) fit these conditions because:
[tex]\[ 10 \times 10 = 100 \][/tex]
[tex]\[ 10 + 10 = 20 \][/tex]
4. Rewrite the factors in terms of \( a \):
[tex]\[ 4a^2 + 20a + 25 = (2a + 5)(2a + 5) \][/tex]
5. Simplify the factors:
[tex]\[ (2a + 5)^2 \][/tex]
### (e) \( 9r^2 + 60r + 100 \)
1. Factor out the quadratic and constant terms.
2. Look for two numbers that multiply to \( 900 \) (i.e., \( 9 \times 100 \)) and add to \( 60 \) (linear term coefficient).
3. The numbers \( 30 \) and \( 30 \) fit these conditions because:
[tex]\[ 30 \times 30 = 900 \][/tex]
[tex]\[ 30 + 30 = 60 \][/tex]
4. Rewrite the factors in terms of \( r \):
[tex]\[ 9r^2 + 60r + 100 = (3r + 10)(3r + 10) \][/tex]
5. Simplify the factors:
[tex]\[ (3r + 10)^2 \][/tex]
### (f) \( 36x^2 + 84x + 49 \)
1. Factor out the quadratic and constant terms.
2. Look for two numbers that multiply to \( 1764 \) (i.e., \( 36 \times 49 \)) and add to \( 84 \) (linear term coefficient).
3. The numbers \( 42 \) and \( 42 \) fit these conditions because:
[tex]\[ 42 \times 42 = 1764 \][/tex]
[tex]\[ 42 + 42 = 84 \][/tex]
4. Rewrite the factors in terms of \( x \):
[tex]\[ 36x^2 + 84x + 49 = (6x + 7)(6x + 7) \][/tex]
5. Simplify the factors:
[tex]\[ (6x + 7)^2 \][/tex]
### (g) \( x^2 - 8x + 16 \)
1. Look for two numbers that multiply to \( 16 \) (the constant term) and add to \( -8 \) (the coefficient of the linear term \( x \)).
2. The numbers \( -4 \) and \( -4 \) fit these conditions because:
[tex]\[ -4 \times -4 = 16 \][/tex]
[tex]\[ -4 + -4 = -8 \][/tex]
3. Rewrite the factors:
[tex]\[ x^2 - 8x + 16 = (x - 4)(x - 4) \][/tex]
4. Simplify the factors:
[tex]\[ (x - 4)^2 \][/tex]
### (h) \( a^2 - 18a + 81 \)
1. Look for two numbers that multiply to \( 81 \) (the constant term) and add to \( -18 \) (the coefficient of the linear term \( a \)).
2. The numbers \( -9 \) and \( -9 \) fit these conditions because:
[tex]\[ -9 \times -9 = 81 \][/tex]
[tex]\[ -9 + -9 = -18 \][/tex]
3. Rewrite the factors:
[tex]\[ a^2 - 18a + 81 = (a - 9)(a - 9) \][/tex]
4. Simplify the factors:
[tex]\[ (a - 9)^2 \][/tex]
### (i) \( p^2 - 26p + 169 \)
1. Look for two numbers that multiply to \( 169 \) (the constant term) and add to \( -26 \) (the coefficient of the linear term \( p \)).
2. The numbers \( -13 \) and \( -13 \) fit these conditions because:
[tex]\[ -13 \times -13 = 169 \][/tex]
[tex]\[ -13 + -13 = -26 \][/tex]
3. Rewrite the factors:
[tex]\[ p^2 - 26p + 169 = (p - 13)(p - 13) \][/tex]
4. Simplify the factors:
[tex]\[ (p - 13)^2 \][/tex]
These factorized forms of the given polynomials are:
- (a) \( (a + 6)^2 \)
- (b) \( (y + 7)^2 \)
- (c) \( (p + 11)^2 \)
- (d) \( (2a + 5)^2 \)
- (e) \( (3r + 10)^2 \)
- (f) \( (6x + 7)^2 \)
- (g) \( (x - 4)^2 \)
- (h) \( (a - 9)^2 \)
- (i) [tex]\( (p - 13)^2 \)[/tex]
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