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Sagot :
To simplify the expression
[tex]\[ \frac{p^2 + pq + q^2}{p + q} - \frac{p^2 - pq + q^2}{p + q}, \][/tex]
we proceed as follows:
1. Combine the fractions over a common denominator:
Both fractions have the same denominator, [tex]\( p + q \)[/tex]. Therefore, we can combine them into a single fraction:
[tex]\[ \frac{(p^2 + pq + q^2) - (p^2 - pq + q^2)}{p + q}. \][/tex]
2. Simplify the numerator by distributing and combining like terms:
Let's expand the numerator:
[tex]\[ (p^2 + pq + q^2) - (p^2 - pq + q^2). \][/tex]
Distribute the negative sign through the terms inside the second parenthesis:
[tex]\[ p^2 + pq + q^2 - p^2 + pq - q^2. \][/tex]
Combine like terms. Notice that [tex]\( p^2 - p^2 \)[/tex] and [tex]\( q^2 - q^2 \)[/tex] cancel out:
[tex]\[ pq + pq = 2pq. \][/tex]
So, the numerator simplifies to [tex]\( 2pq \)[/tex].
3. Form the simplified fraction:
Now, we have the simplified numerator over the common denominator:
[tex]\[ \frac{2pq}{p + q}. \][/tex]
Therefore, the expression simplifies to:
[tex]\[ \frac{2pq}{p + q}. \][/tex]
[tex]\[ \frac{p^2 + pq + q^2}{p + q} - \frac{p^2 - pq + q^2}{p + q}, \][/tex]
we proceed as follows:
1. Combine the fractions over a common denominator:
Both fractions have the same denominator, [tex]\( p + q \)[/tex]. Therefore, we can combine them into a single fraction:
[tex]\[ \frac{(p^2 + pq + q^2) - (p^2 - pq + q^2)}{p + q}. \][/tex]
2. Simplify the numerator by distributing and combining like terms:
Let's expand the numerator:
[tex]\[ (p^2 + pq + q^2) - (p^2 - pq + q^2). \][/tex]
Distribute the negative sign through the terms inside the second parenthesis:
[tex]\[ p^2 + pq + q^2 - p^2 + pq - q^2. \][/tex]
Combine like terms. Notice that [tex]\( p^2 - p^2 \)[/tex] and [tex]\( q^2 - q^2 \)[/tex] cancel out:
[tex]\[ pq + pq = 2pq. \][/tex]
So, the numerator simplifies to [tex]\( 2pq \)[/tex].
3. Form the simplified fraction:
Now, we have the simplified numerator over the common denominator:
[tex]\[ \frac{2pq}{p + q}. \][/tex]
Therefore, the expression simplifies to:
[tex]\[ \frac{2pq}{p + q}. \][/tex]
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